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integer representing the index of the matching parenthesis. If no parenthesis matches nothing is returned
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Integer representing the index of the matching parenthesis. If no parenthesis matches, return ``None``.
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def associated_parenthesis(self, pos): r""" report the position for the parenthesis that matches the one at position ``pos``
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Bijection of Biane from Dyck words to non crossing partitions
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Bijection of Biane from Dyck words to non-crossing partitions.
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def to_noncrossing_partition(self): r""" Bijection of Biane from Dyck words to non crossing partitions Thanks to Mathieu Dutour for describing the bijection. EXAMPLES:: sage: DyckWord([]).to_noncrossing_partition() [] sage: DyckWord([1, 0]).to_noncrossing_partition() [[1]] sage: DyckWord([1, 1, 0, 0]).to_noncrossing_partition() [[1, 2]] sage: DyckWord([1, 1, 1, 0, 0, 0]).to_noncrossing_partition() [[1, 2, 3]] sage: DyckWord([1, 0, 1, 0, 1, 0]).to_noncrossing_partition() [[1], [2], [3]] sage: DyckWord([1, 1, 0, 1, 0, 0]).to_noncrossing_partition() [[2], [1, 3]] """ partition = [] stack = [] i = 0 p = 1
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returns a standard tableau of length less than or equal to 2 with the size the same as the length of the list the standard tableau will be rectangular iff ``self`` is a complete Dyck word
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Returns a standard tableau of length less than or equal to 2 with the size the same as the length of the list. The standard tableau will be rectangular iff ``self`` is a complete Dyck word.
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def to_tableau(self): r""" returns a standard tableau of length less than or equal to 2 with the size the same as the length of the list the standard tableau will be rectangular iff ``self`` is a complete Dyck word
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TODO: better name? to_standard_tableau? and should *actually* return a Tableau object?
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TODO: better name? ``to_standard_tableau``? and should *actually* return a Tableau object?
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def to_tableau(self): r""" returns a standard tableau of length less than or equal to 2 with the size the same as the length of the list the standard tableau will be rectangular iff ``self`` is a complete Dyck word
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Returns the a-statistic for the Dyck word correspond to the area of the Dyck path.
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Returns the a-statistic for the Dyck word corresponding to the area of the Dyck path.
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def a_statistic(self): """ Returns the a-statistic for the Dyck word correspond to the area of the Dyck path. One can view a balanced Dyck word as a lattice path from `(0,0)` to `(n,n)` in the first quadrant by letting '1's represent steps in the direction `(1,0)` and '0's represent steps in the direction `(0,1)`. The resulting path will remain weakly above the diagonal `y = x`. The a-statistic, or area statistic, is the number of complete squares in the integer lattice which are below the path and above the line `y = x`. The 'half-squares' directly above the line `y=x` do not contribute to this statistic. EXAMPLES:: sage: dw = DyckWord([1,0,1,0]) sage: dw.a_statistic() # 2 half-squares, 0 complete squares 0 :: sage: dw = DyckWord([1,1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0,0]) sage: dw.a_statistic() 19 :: sage: DyckWord([1,1,1,1,0,0,0,0]).a_statistic() 6 sage: DyckWord([1,1,1,0,1,0,0,0]).a_statistic() 5 sage: DyckWord([1,1,1,0,0,1,0,0]).a_statistic() 4 sage: DyckWord([1,1,1,0,0,0,1,0]).a_statistic() 3 sage: DyckWord([1,0,1,1,0,1,0,0]).a_statistic() 2 sage: DyckWord([1,1,0,1,1,0,0,0]).a_statistic() 4 sage: DyckWord([1,1,0,0,1,1,0,0]).a_statistic() 2 sage: DyckWord([1,0,1,1,1,0,0,0]).a_statistic() 3 sage: DyckWord([1,0,1,1,0,0,1,0]).a_statistic() 1 sage: DyckWord([1,0,1,0,1,1,0,0]).a_statistic() 1 sage: DyckWord([1,1,0,0,1,0,1,0]).a_statistic() 1 sage: DyckWord([1,1,0,1,0,0,1,0]).a_statistic() 2 sage: DyckWord([1,1,0,1,0,1,0,0]).a_statistic() 3 sage: DyckWord([1,0,1,0,1,0,1,0]).a_statistic() 0 """ above = 0 diagonal = 0 a = 0 for move in self: if move == 1: above += 1 elif move == 0: diagonal += 1 a += above - diagonal return a
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The bouncing ball will strike the diagonal at places $(0, 0), (j_1, j_1), (j_2, j_2), ... , (j_r-1, j_r-1), (j_r, j_r) = (n, n).$
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The bouncing ball will strike the diagonal at places .. MATH:: (0, 0), (j_1, j_1), (j_2, j_2), \dots , (j_r-1, j_r-1), (j_r, j_r) = (n, n).
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def b_statistic(self): r""" Returns the b-statistic for the Dyck word corresponding to the bounce statistic of the Dyck word. One can view a balanced Dyck word as a lattice path from `(0,0)` to `(n,n)` in the first quadrant by letting '1's represent steps in the direction `(0,1)` and '0's represent steps in the direction `(1,0)`. The resulting path will remain weakly above the diagonal `y = x`. We describe the b-statistic of such a path in terms of what is known as the "bounce path". We can think of our bounce path as describing the trail of a billiard ball shot West from (n, n), which "bounces" down whenever it encounters a vertical step and "bounces" left when it encounters the line y = x.
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converts a non-crossing partition to a Dyck word
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Converts a non-crossing partition to a Dyck word.
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def from_noncrossing_partition(ncp): r""" converts a non-crossing partition to a Dyck word TESTS:: sage: DyckWord(noncrossing_partition=[[1,2]]) # indirect doctest [1, 1, 0, 0] sage: DyckWord(noncrossing_partition=[[1],[2]]) [1, 0, 1, 0] :: sage: dws = DyckWords(5).list() sage: ncps = map( lambda x: x.to_noncrossing_partition(), dws) sage: dws2 = map( lambda x: DyckWord(noncrossing_partition=x), ncps) sage: dws == dws2 True """ l = [ 0 ] * int( sum( [ len(v) for v in ncp ] ) ) for v in ncp: l[v[-1]-1] = len(v) res = [] for i in l: res += [ open_symbol ] + [close_symbol]*int(i) return DyckWord(res)
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Mtrans = Matrix(k, 2, M2)*M1inv
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Mtrans = Matrix(k, 2, M2)*Maux*M1inv assert Mtrans[1][0] in N
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def is_Gamma0_equivalent(self, other, N, Transformation=False): r""" Checks if cusps ``self`` and ``other`` are `\Gamma_0(N)`- equivalent.
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and only if `p_2` dominates `p_1`.
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and only if the conjugate of `p_2` dominates `p_1`.
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def gale_ryser_theorem(p1, p2, algorithm="ryser"): r""" Returns the binary matrix given by the Gale-Ryser theorem. The Gale Ryser theorem asserts that if `p_1,p_2` are two partitions of `n` of respective lengths `k_1,k_2`, then there is a binary `k_1\times k_2` matrix `M` such that `p_1` is the vector of row sums and `p_2` is the vector of column sums of `M`, if and only if `p_2` dominates `p_1`. INPUT: - ``p1`` -- the first partition of `n` (trailing 0's allowed) - ``p2`` -- the second partition of `n` (trailing 0's allowed) - ``algorithm`` -- two possible string values : - ``"ryser"`` (default) implements the construction due to Ryser [Ryser63]_. - ``"gale"`` implements the construction due to Gale [Gale57]_. OUTPUT: - A binary matrix if it exists, ``None`` otherwise. Gale's Algorithm: (Gale [Gale57]_): A matrix satisfying the constraints of its sums can be defined as the solution of the following Linear Program, which Sage knows how to solve (requires packages GLPK or CBC). .. MATH:: \forall i&\sum_{j=1}^{k_2} b_{i,j}=p_{1,j}\\ \forall i&\sum_{j=1}^{k_1} b_{j,i}=p_{2,j}\\ &b_{i,j}\mbox{ is a binary variable} Ryser's Algorithm: (Ryser [Ryser63]_): The construction of an `m\times n` matrix `A=A_{r,s}`, due to Ryser, is described as follows. The construction works if and only if have `s\preceq r^*`. * Construct the `m\times n` matrix `B` from `r` by defining the `i`-th row of `B` to be the vector whose first `r_i` entries are `1`, and the remainder are 0's, `1\leq i\leq m`. This maximal matrix `B` with row sum `r` and ones left justified has column sum `r^{*}`. * Shift the last `1` in certain rows of `B` to column `n` in order to achieve the sum `s_n`. Call this `B` again. * The `1`'s in column n are to appear in those rows in which `A` has the largest row sums, giving preference to the bottom-most positions in case of ties. * Note: When this step automatically "fixes" other columns, one must skip ahead to the first column index with a wrong sum in the step below. * Proceed inductively to construct columns `n-1`, ..., `2`, `1`. * Set `A = B`. Return `A`. EXAMPLES: Computing the matrix for `p_1=p_2=2+2+1` :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [2,2,1] sage: p2 = [2,2,1] sage: print gale_ryser_theorem(p1, p2, algorithm="gale") # Optional - requires GLPK or CBC [0 1 1] [1 1 0] [1 0 0] Or for a non-square matrix with `p_1=3+3+2+1` and `p_2=3+2+2+1+1` :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [3,3,1,1] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2) [1 1 1 0] [1 1 0 1] [1 0 0 0] [0 1 0 0] sage: p1 = [4,2,2] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2) [1 1 1 1] [1 1 0 0] [1 1 0 0] sage: p1 = [4,2,2,0] sage: p2 = [3,3,1,1,0,0] sage: gale_ryser_theorem(p1, p2) [1 1 1 1 0 0] [1 1 0 0 0 0] [1 1 0 0 0 0] [0 0 0 0 0 0] sage: p1 = [3,3,2,1] sage: p2 = [3,2,2,1,1] sage: print gale_ryser_theorem(p1, p2, algorithm="gale") # Optional - requires GLPK or CBC [1 0 1 1 0] [1 0 1 0 1] [1 1 0 0 0] [0 1 0 0 0] With `0` in the sequences, and with unordered inputs :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: gale_ryser_theorem([3,3,0,1,1,0], [3,1,3,1,0]) [1 1 1 0 0] [1 0 1 1 0] [0 0 0 0 0] [1 0 0 0 0] [0 0 1 0 0] [0 0 0 0 0] REFERENCES: .. [Ryser63] H. J. Ryser, Combinatorial Mathematics, Carus Monographs, MAA, 1963. .. [Gale57] D. Gale, A theorem on flows in networks, Pacific J. Math. 7(1957)1073-1082. """ from sage.combinat.partition import Partition from sage.matrix.constructor import matrix if not(is_gale_ryser(p1,p2)): return False if algorithm=="ryser": # ryser's algorithm from sage.combinat.permutation import Permutation # Sorts the sequences if they are not, and remembers the permutation # applied tmp = sorted(enumerate(p1), reverse=True, key=lambda x:x[1]) r = [x[1] for x in tmp if x[1]>0] r_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()] m = len(r) tmp = sorted(enumerate(p2), reverse=True, key=lambda x:x[1]) s = [x[1] for x in tmp if x[1]>0] s_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()] n = len(s) rowsA0 = [[0]*n]*m for j in range(m): if j<m: rowsA0[j] = [1]*r[j]+[0]*(n-r[j]) else: rowsA0[j] = [0]*n A0 = matrix(rowsA0) for j in range(1,n-1): # starts for loop, k = n-1, ..., 1 # which finds the 1st column with # incorrect column sum. For that bad # column index, apply slider again for k in range(1,n): if sum(A0.column(n-k))<>s[n-k]: break A0 = _slider01(A0,s[n-k],n-k) # If we need to add empty rows/columns if len(p1)!=m: A0 = A0.stack(matrix([[0]*n]*(len(p1)-m))) if len(p2)!=n: A0 = A0.transpose().stack(matrix([[0]*len(p1)]*(len(p2)-n))).transpose() # Applying the permutations to get a matrix satisfying the # order given by the input A0 = A0.matrix_from_rows_and_columns(r_permutation, s_permutation) return A0 elif algorithm == "gale": from sage.numerical.mip import MixedIntegerLinearProgram k1, k2=len(p1), len(p2) p = MixedIntegerLinearProgram() b = p.new_variable(dim=2) for (i,c) in enumerate(p1): p.add_constraint(sum([b[i][j] for j in xrange(k2)]),min=c,max=c) for (i,c) in enumerate(p2): p.add_constraint(sum([b[j][i] for j in xrange(k1)]),min=c,max=c) p.set_objective(None) p.set_binary(b) p.solve() b = p.get_values(b) M = [[0]*k2 for i in xrange(k1)] for i in xrange(k1): for j in xrange(k2): M[i][j] = int(b[i][j]) return matrix(M) else: raise ValueError("The only two algorithms available are \"gale\" and \"ryser\"")
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- ``p1`` -- the first partition of `n` (trailing 0's allowed) - ``p2`` -- the second partition of `n` (trailing 0's allowed)
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- ``p1, p2``-- list of integers representing the vectors of row/column sums
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def gale_ryser_theorem(p1, p2, algorithm="ryser"): r""" Returns the binary matrix given by the Gale-Ryser theorem. The Gale Ryser theorem asserts that if `p_1,p_2` are two partitions of `n` of respective lengths `k_1,k_2`, then there is a binary `k_1\times k_2` matrix `M` such that `p_1` is the vector of row sums and `p_2` is the vector of column sums of `M`, if and only if `p_2` dominates `p_1`. INPUT: - ``p1`` -- the first partition of `n` (trailing 0's allowed) - ``p2`` -- the second partition of `n` (trailing 0's allowed) - ``algorithm`` -- two possible string values : - ``"ryser"`` (default) implements the construction due to Ryser [Ryser63]_. - ``"gale"`` implements the construction due to Gale [Gale57]_. OUTPUT: - A binary matrix if it exists, ``None`` otherwise. Gale's Algorithm: (Gale [Gale57]_): A matrix satisfying the constraints of its sums can be defined as the solution of the following Linear Program, which Sage knows how to solve (requires packages GLPK or CBC). .. MATH:: \forall i&\sum_{j=1}^{k_2} b_{i,j}=p_{1,j}\\ \forall i&\sum_{j=1}^{k_1} b_{j,i}=p_{2,j}\\ &b_{i,j}\mbox{ is a binary variable} Ryser's Algorithm: (Ryser [Ryser63]_): The construction of an `m\times n` matrix `A=A_{r,s}`, due to Ryser, is described as follows. The construction works if and only if have `s\preceq r^*`. * Construct the `m\times n` matrix `B` from `r` by defining the `i`-th row of `B` to be the vector whose first `r_i` entries are `1`, and the remainder are 0's, `1\leq i\leq m`. This maximal matrix `B` with row sum `r` and ones left justified has column sum `r^{*}`. * Shift the last `1` in certain rows of `B` to column `n` in order to achieve the sum `s_n`. Call this `B` again. * The `1`'s in column n are to appear in those rows in which `A` has the largest row sums, giving preference to the bottom-most positions in case of ties. * Note: When this step automatically "fixes" other columns, one must skip ahead to the first column index with a wrong sum in the step below. * Proceed inductively to construct columns `n-1`, ..., `2`, `1`. * Set `A = B`. Return `A`. EXAMPLES: Computing the matrix for `p_1=p_2=2+2+1` :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [2,2,1] sage: p2 = [2,2,1] sage: print gale_ryser_theorem(p1, p2, algorithm="gale") # Optional - requires GLPK or CBC [0 1 1] [1 1 0] [1 0 0] Or for a non-square matrix with `p_1=3+3+2+1` and `p_2=3+2+2+1+1` :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [3,3,1,1] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2) [1 1 1 0] [1 1 0 1] [1 0 0 0] [0 1 0 0] sage: p1 = [4,2,2] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2) [1 1 1 1] [1 1 0 0] [1 1 0 0] sage: p1 = [4,2,2,0] sage: p2 = [3,3,1,1,0,0] sage: gale_ryser_theorem(p1, p2) [1 1 1 1 0 0] [1 1 0 0 0 0] [1 1 0 0 0 0] [0 0 0 0 0 0] sage: p1 = [3,3,2,1] sage: p2 = [3,2,2,1,1] sage: print gale_ryser_theorem(p1, p2, algorithm="gale") # Optional - requires GLPK or CBC [1 0 1 1 0] [1 0 1 0 1] [1 1 0 0 0] [0 1 0 0 0] With `0` in the sequences, and with unordered inputs :: sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: gale_ryser_theorem([3,3,0,1,1,0], [3,1,3,1,0]) [1 1 1 0 0] [1 0 1 1 0] [0 0 0 0 0] [1 0 0 0 0] [0 0 1 0 0] [0 0 0 0 0] REFERENCES: .. [Ryser63] H. J. Ryser, Combinatorial Mathematics, Carus Monographs, MAA, 1963. .. [Gale57] D. Gale, A theorem on flows in networks, Pacific J. Math. 7(1957)1073-1082. """ from sage.combinat.partition import Partition from sage.matrix.constructor import matrix if not(is_gale_ryser(p1,p2)): return False if algorithm=="ryser": # ryser's algorithm from sage.combinat.permutation import Permutation # Sorts the sequences if they are not, and remembers the permutation # applied tmp = sorted(enumerate(p1), reverse=True, key=lambda x:x[1]) r = [x[1] for x in tmp if x[1]>0] r_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()] m = len(r) tmp = sorted(enumerate(p2), reverse=True, key=lambda x:x[1]) s = [x[1] for x in tmp if x[1]>0] s_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()] n = len(s) rowsA0 = [[0]*n]*m for j in range(m): if j<m: rowsA0[j] = [1]*r[j]+[0]*(n-r[j]) else: rowsA0[j] = [0]*n A0 = matrix(rowsA0) for j in range(1,n-1): # starts for loop, k = n-1, ..., 1 # which finds the 1st column with # incorrect column sum. For that bad # column index, apply slider again for k in range(1,n): if sum(A0.column(n-k))<>s[n-k]: break A0 = _slider01(A0,s[n-k],n-k) # If we need to add empty rows/columns if len(p1)!=m: A0 = A0.stack(matrix([[0]*n]*(len(p1)-m))) if len(p2)!=n: A0 = A0.transpose().stack(matrix([[0]*len(p1)]*(len(p2)-n))).transpose() # Applying the permutations to get a matrix satisfying the # order given by the input A0 = A0.matrix_from_rows_and_columns(r_permutation, s_permutation) return A0 elif algorithm == "gale": from sage.numerical.mip import MixedIntegerLinearProgram k1, k2=len(p1), len(p2) p = MixedIntegerLinearProgram() b = p.new_variable(dim=2) for (i,c) in enumerate(p1): p.add_constraint(sum([b[i][j] for j in xrange(k2)]),min=c,max=c) for (i,c) in enumerate(p2): p.add_constraint(sum([b[j][i] for j in xrange(k1)]),min=c,max=c) p.set_objective(None) p.set_binary(b) p.solve() b = p.get_values(b) M = [[0]*k2 for i in xrange(k1)] for i in xrange(k1): for j in xrange(k2): M[i][j] = int(b[i][j]) return matrix(M) else: raise ValueError("The only two algorithms available are \"gale\" and \"ryser\"")
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This equation can be solved within Maxima but not within Sage. It needs assumptions assume(x>0,y>0) and works in Maxima, but not in Sage:: sage: assume(x>0) sage: assume(y>0) sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y,y,show_method=True)
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def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): r""" Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependent variable (hereafter called ``y``) - ``ics`` - (optional) the initial or boundary conditions - for a first-order equation, specify the initial ``x`` and ``y`` - for a second-order equation, specify the initial ``x``, ``y``, and ``dy/dx``, i.e. write `[x_0, y(x_0), y'(x_0)]` - for a second-order boundary solution, specify initial and final ``x`` and ``y`` boundary conditions, i.e. write `[x_0, y(x_0), x_1, y(x_1)]`. - gives an error if the solution is not SymbolicEquation (as happens for example for Clairaut equation) - ``ivar`` - (optional) the independent variable (hereafter called x), which must be specified if there is more than one independent variable in the equation. - ``show_method`` - (optional) if true, then Sage returns pair ``[solution, method]``, where method is the string describing method which has been used to get solution (Maxima uses the following order for first order equations: linear, separable, exact (including exact with integrating factor), homogeneous, bernoulli, generalized homogeneous) - use carefully in class, see below for the example of the equation which is separable but this property is not recognized by Maxima and equation is solved as exact. - ``contrib_ode`` - (optional) if true, desolve allows to solve clairaut, lagrange, riccati and some other equations. May take a long time and thus turned off by default. Initial conditions can be used only if the result is one SymbolicEquation (does not contain singular solution, for example) OUTPUT: In most cases returns SymbolicEquation which defines the solution implicitly. If the result is in the form y(x)=... (happens for linear eqs.), returns the right-hand side only. The possible constant solutions of separable ODE's are omitted. EXAMPLES:: sage: x = var('x') sage: y = function('y', x) sage: desolve(diff(y,x) + y - 1, y) (c + e^x)*e^(-x) :: sage: f = desolve(diff(y,x) + y - 1, y, ics=[10,2]); f (e^10 + e^x)*e^(-x) :: sage: plot(f) We can also solve second-order differential equations.:: sage: x = var('x') sage: y = function('y', x) sage: de = diff(y,x,2) - y == x sage: desolve(de, y) k1*e^x + k2*e^(-x) - x :: sage: f = desolve(de, y, [10,2,1]); f -x + 5*e^(-x + 10) + 7*e^(x - 10) :: sage: f(x=10) 2 :: sage: diff(f,x)(x=10) 1 :: sage: de = diff(y,x,2) + y == 0 sage: desolve(de, y) k1*sin(x) + k2*cos(x) :: sage: desolve(de, y, [0,1,pi/2,4]) 4*sin(x) + cos(x) :: sage: desolve(y*diff(y,x)+sin(x)==0,y) -1/2*y(x)^2 == c - cos(x) Clairot equation: general and singular solutions:: sage: desolve(diff(y,x)^2+x*diff(y,x)-y==0,y,contrib_ode=True,show_method=True) [[y(x) == c^2 + c*x, y(x) == -1/4*x^2], 'clairault'] For equations involving more variables we specify independent variable:: sage: a,b,c,n=var('a b c n') sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True) [[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]] :: sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True,show_method=True) [[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]], 'riccati'] Higher orded, not involving independent variable:: sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y).expand() 1/6*y(x)^3 + k1*y(x) == k2 + x :: sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3]).expand() 1/6*y(x)^3 - 5/3*y(x) == x - 3/2 :: sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3],show_method=True) [1/6*y(x)^3 - 5/3*y(x) == x - 3/2, 'freeofx'] Separable equations - Sage returns solution in implicit form:: sage: desolve(diff(y,x)*sin(y) == cos(x),y) -cos(y(x)) == c + sin(x) :: sage: desolve(diff(y,x)*sin(y) == cos(x),y,show_method=True) [-cos(y(x)) == c + sin(x), 'separable'] :: sage: desolve(diff(y,x)*sin(y) == cos(x),y,[pi/2,1]) -cos(y(x)) == sin(x) - cos(1) - 1 Linear equation - Sage returns the expression on the right hand side only:: sage: desolve(diff(y,x)+(y) == cos(x),y) 1/2*((sin(x) + cos(x))*e^x + 2*c)*e^(-x) :: sage: desolve(diff(y,x)+(y) == cos(x),y,show_method=True) [1/2*((sin(x) + cos(x))*e^x + 2*c)*e^(-x), 'linear'] :: sage: desolve(diff(y,x)+(y) == cos(x),y,[0,1]) 1/2*(e^x*sin(x) + e^x*cos(x) + 1)*e^(-x) This ODE with separated variables is solved as exact. Explanation - factor does not split `e^{x-y}` in Maxima into `e^{x}e^{y}`:: sage: desolve(diff(y,x)==exp(x-y),y,show_method=True) [-e^x + e^y(x) == c, 'exact'] You can solve Bessel equations. You can also use initial conditions, but you cannot put (sometimes desired) initial condition at x=0, since this point is singlar point of the equation. Anyway, if the solution should be bounded at x=0, then k2=0.:: sage: desolve(x^2*diff(y,x,x)+x*diff(y,x)+(x^2-4)*y==0,y) k1*bessel_j(2, x) + k2*bessel_y(2, x) Difficult ODE produces error:: sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y) # not tested Traceback (click to the left for traceback) ... NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True." Difficult ODE produces error - moreover, takes a long time :: sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y,contrib_ode=True) # not tested Some more types od ODE's:: sage: desolve(x*diff(y,x)^2-(1+x*y)*diff(y,x)+y==0,y,contrib_ode=True,show_method=True) [[y(x) == c + log(x), y(x) == c*e^x], 'factor'] :: sage: desolve(diff(y,x)==(x+y)^2,y,contrib_ode=True,show_method=True) [[[x == c - arctan(sqrt(t)), y(x) == -x - sqrt(t)], [x == c + arctan(sqrt(t)), y(x) == -x + sqrt(t)]], 'lagrange'] These two examples produce error (as expected, Maxima 5.18 cannot solve equations from initial conditions). Current Maxima 5.18 returns false answer in this case!:: sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2]).expand() # not tested Traceback (click to the left for traceback) ... NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True." :: sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2],show_method=True) # not tested Traceback (click to the left for traceback) ... NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True." Second order linear ODE:: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y) (k2*x + k1)*e^(-x) + 1/2*sin(x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,show_method=True) [(k2*x + k1)*e^(-x) + 1/2*sin(x), 'variationofparameters'] :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1]) 1/2*(7*x + 6)*e^(-x) + 1/2*sin(x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1],show_method=True) [1/2*(7*x + 6)*e^(-x) + 1/2*sin(x), 'variationofparameters'] :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2]) 3*((e^(1/2*pi) - 2)*x/pi + 1)*e^(-x) + 1/2*sin(x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2],show_method=True) [3*((e^(1/2*pi) - 2)*x/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters'] :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y) (k2*x + k1)*e^(-x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,show_method=True) [(k2*x + k1)*e^(-x), 'constcoeff'] :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1]) (4*x + 3)*e^(-x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1],show_method=True) [(4*x + 3)*e^(-x), 'constcoeff'] :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2]) (2*(2*e^(1/2*pi) - 3)*x/pi + 3)*e^(-x) :: sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2],show_method=True) [(2*(2*e^(1/2*pi) - 3)*x/pi + 3)*e^(-x), 'constcoeff'] This equation can be solved within Maxima but not within Sage. It needs assumptions assume(x>0,y>0) and works in Maxima, but not in Sage:: sage: assume(x>0) # not tested sage: assume(y>0) # not tested sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y,y,show_method=True) # not tested TESTS: Trac #6479 fixed:: sage: x = var('x') sage: y = function('y', x) sage: desolve( diff(y,x,x) == 0, y, [0,0,1]) x :: sage: desolve( diff(y,x,x) == 0, y, [0,1,1]) x + 1 Trac #9835 fixed:: sage: x = var('x') sage: y = function('y', x) sage: desolve(diff(y,x,2)+y*(1-y^2)==0,y,[0,-1,1,1]) Traceback (most recent call last): ... NotImplementedError: Unable to use initial condition for this equation (freeofx). AUTHORS: - David Joyner (1-2006) - Robert Bradshaw (10-2008) - Robert Marik (10-2009) """ if is_SymbolicEquation(de): de = de.lhs() - de.rhs() if is_SymbolicVariable(dvar): raise ValueError, "You have to declare dependent variable as a function, eg. y=function('y',x)" # for backwards compatibility if isinstance(dvar, list): dvar, ivar = dvar elif ivar is None: ivars = de.variables() ivars = [t for t in ivars if t is not dvar] if len(ivars) != 1: raise ValueError, "Unable to determine independent variable, please specify." ivar = ivars[0] def sanitize_var(exprs): return exprs.replace("'"+dvar_str+"("+ivar_str+")",dvar_str) de00 = de._maxima_() P = de00.parent() dvar_str=P(dvar.operator()).str() ivar_str=P(ivar).str() de00 = de00.str() de0 = sanitize_var(de00) ode_solver="ode2" cmd="(TEMP:%s(%s,%s,%s), if TEMP=false then TEMP else substitute(%s=%s(%s),TEMP))"%(ode_solver,de0,dvar_str,ivar_str,dvar_str,dvar_str,ivar_str) # we produce string like this # ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y(x),x) soln = P(cmd) if str(soln).strip() == 'false': if contrib_ode: ode_solver="contrib_ode" P("load('contrib_ode)") cmd="(TEMP:%s(%s,%s,%s), if TEMP=false then TEMP else substitute(%s=%s(%s),TEMP))"%(ode_solver,de0,dvar_str,ivar_str,dvar_str,dvar_str,ivar_str) # we produce string like this # (TEMP:contrib_ode(x*('diff(y,x,1))^2-(x*y+1)*'diff(y,x,1)+y,y,x), if TEMP=false then TEMP else substitute(y=y(x),TEMP)) soln = P(cmd) if str(soln).strip() == 'false': raise NotImplementedError, "Maxima was unable to solve this ODE." else: raise NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True." if show_method: maxima_method=P("method") if (ics is not None): if not is_SymbolicEquation(soln.sage()): if not show_method: maxima_method=P("method") raise NotImplementedError, "Unable to use initial condition for this equation (%s)."%(str(maxima_method).strip()) if len(ics) == 2: tempic=(ivar==ics[0])._maxima_().str() tempic=tempic+","+(dvar==ics[1])._maxima_().str() cmd="(TEMP:ic1(%s(%s,%s,%s),%s),substitute(%s=%s(%s),TEMP))"%(ode_solver,de00,dvar_str,ivar_str,tempic,dvar_str,dvar_str,ivar_str) cmd=sanitize_var(cmd) # we produce string like this # (TEMP:ic2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,'diff(y,x)=1),substitute(y=y(x),TEMP)) soln=P(cmd) if len(ics) == 3: #fixed ic2 command from Maxima - we have to ensure that %k1, %k2 do not depend on variables, should be removed when fixed in Maxima P("ic2_sage(soln,xa,ya,dya):=block([programmode:true,backsubst:true,singsolve:true,temp,%k2,%k1,TEMP_k], \ noteqn(xa), noteqn(ya), noteqn(dya), boundtest('%k1,%k1), boundtest('%k2,%k2), \ temp: lhs(soln) - rhs(soln), \ TEMP_k:solve([subst([xa,ya],soln), subst([dya,xa], lhs(dya)=-subst(0,lhs(dya),diff(temp,lhs(xa)))/diff(temp,lhs(ya)))],[%k1,%k2]), \ if not freeof(lhs(ya),TEMP_k) or not freeof(lhs(xa),TEMP_k) then return (false), \ temp: maplist(lambda([zz], subst(zz,soln)), TEMP_k), \ if length(temp)=1 then return(first(temp)) else return(temp))") tempic=P(ivar==ics[0]).str() tempic=tempic+","+P(dvar==ics[1]).str() tempic=tempic+",'diff("+dvar_str+","+ivar_str+")="+P(ics[2]).str() cmd="(TEMP:ic2_sage(%s(%s,%s,%s),%s),substitute(%s=%s(%s),TEMP))"%(ode_solver,de00,dvar_str,ivar_str,tempic,dvar_str,dvar_str,ivar_str) cmd=sanitize_var(cmd) # we produce string like this # (TEMP:ic2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,'diff(y,x)=1),substitute(y=y(x),TEMP)) soln=P(cmd) if str(soln).strip() == 'false': raise NotImplementedError, "Maxima was unable to solve this IVP. Remove the initial condition to get the general solution." if len(ics) == 4: #fixed bc2 command from Maxima - we have to ensure that %k1, %k2 do not depend on variables, should be removed when fixed in Maxima P("bc2_sage(soln,xa,ya,xb,yb):=block([programmode:true,backsubst:true,singsolve:true,temp,%k1,%k2,TEMP_k], \ noteqn(xa), noteqn(ya), noteqn(xb), noteqn(yb), boundtest('%k1,%k1), boundtest('%k2,%k2), \ TEMP_k:solve([subst([xa,ya],soln), subst([xb,yb],soln)], [%k1,%k2]), \ if not freeof(lhs(ya),TEMP_k) or not freeof(lhs(xa),TEMP_k) then return (false), \ temp: maplist(lambda([zz], subst(zz,soln)),TEMP_k), \ if length(temp)=1 then return(first(temp)) else return(temp))") cmd="bc2_sage(%s(%s,%s,%s),%s,%s=%s,%s,%s=%s)"%(ode_solver,de00,dvar_str,ivar_str,P(ivar==ics[0]).str(),dvar_str,P(ics[1]).str(),P(ivar==ics[2]).str(),dvar_str,P(ics[3]).str()) cmd="(TEMP:%s,substitute(%s=%s(%s),TEMP))"%(cmd,dvar_str,dvar_str,ivar_str) cmd=sanitize_var(cmd) # we produce string like this # (TEMP:bc2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,x=%pi/2,y=2),substitute(y=y(x),TEMP)) soln=P(cmd) if str(soln).strip() == 'false': raise NotImplementedError, "Maxima was unable to solve this BVP. Remove the initial condition to get the general solution." soln=soln.sage() if is_SymbolicEquation(soln) and soln.lhs() == dvar: # Remark: Here we do not check that the right hand side does not depend on dvar. # This probably will not hapen for soutions obtained via ode2, anyway. soln = soln.rhs() if show_method: return [soln,maxima_method.str()] else: return soln
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""" c = [[x+1] for x in range(n)] c[n-1] = [] return LatticePoset(c)
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TESTS: Check that sage: Posets.ChainPoset(0) Finite lattice containing 0 elements sage: C = Posets.ChainPoset(1); C Finite lattice containing 1 elements sage: C.cover_relations() [] sage: C = Posets.ChainPoset(2); C Finite lattice containing 2 elements sage: C.cover_relations() [[0, 1]] """ return LatticePoset((range(n), [[x,x+1] for x in range(n-1)]))
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def ChainPoset(self, n): """ Returns a chain (a totally ordered poset) containing ``n`` elements.
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""" c = [[] for x in range(n)] return Poset(c)
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TESTS: Check that sage: Posets.AntichainPoset(0) Finite poset containing 0 elements sage: C = Posets.AntichainPoset(1); C Finite poset containing 1 elements sage: C.cover_relations() [] sage: C = Posets.AntichainPoset(2); C Finite poset containing 2 elements sage: C.cover_relations() [] """ return Poset((range(n), []))
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def AntichainPoset(self, n): """ Returns an antichain (a poset with no comparable elements) containing ``n`` elements.
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self.__modulus = ntl_ZZ_pEContext(ntl_ZZ_pX(list(base_ring.polynomial()), p))
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self._modulus = ntl_ZZ_pEContext(ntl_ZZ_pX(list(base_ring.polynomial()), p))
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def __init__(self, base_ring, name="x", sparse=False, element_class=None, implementation=None): """ TESTS: sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_field as PRing sage: R = PRing(QQ, 'x'); R Univariate Polynomial Ring in x over Rational Field sage: type(R.gen()) <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_rational_dense'> sage: R = PRing(QQ, 'x', sparse=True); R Sparse Univariate Polynomial Ring in x over Rational Field sage: type(R.gen()) <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_field'> sage: R = PRing(CC, 'x'); R Univariate Polynomial Ring in x over Complex Field with 53 bits of precision sage: type(R.gen()) <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'> """ if implementation is None: implementation="NTL" if implementation == "NTL" and \ sage.rings.finite_field.is_FiniteField(base_ring): p=base_ring.characteristic() from sage.libs.ntl.ntl_ZZ_pEContext import ntl_ZZ_pEContext from sage.libs.ntl.ntl_ZZ_pX import ntl_ZZ_pX self.__modulus = ntl_ZZ_pEContext(ntl_ZZ_pX(list(base_ring.polynomial()), p)) from sage.rings.polynomial.polynomial_zz_pex import Polynomial_ZZ_pEX element_class=Polynomial_ZZ_pEX
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P1 = P[0]; P2 = P[1]
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P1 = P[0] P2 = P[1] if is_Integer(P1) and is_Integer(P2): R = PolynomialRing(self.value_ring(), 'x') P1 = R(P1) P2 = R(P2) return JacobianMorphism_divisor_class_field(self, tuple([P1,P2])) if is_Integer(P1) and is_Polynomial(P2): R = PolynomialRing(self.value_ring(), 'x') P1 = R(P1) return JacobianMorphism_divisor_class_field(self, tuple([P1,P2])) if is_Integer(P2) and is_Polynomial(P1): R = PolynomialRing(self.value_ring(), 'x') P2 = R(P2) return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
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def __call__(self, P): r""" Returns a rational point P in the abstract Homset J(K), given: 0. A point P in J = Jac(C), returning P; 1. A point P on the curve C such that J = Jac(C), where C is an odd degree model, returning [P - oo]; 2. A pair of points (P, Q) on the curve C such that J = Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that `b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))]. EXAMPLES:: sage: P.<x> = PolynomialRing(QQ) sage: f = x^5 - x + 1; h = x sage: C = HyperellipticCurve(f,h,'u,v') sage: P = C(0,1,1) sage: J = C.jacobian() sage: Q = J(QQ)(P) sage: for i in range(6): i*Q (1) (u, v - 1) (u^2, v + u - 1) (u^2, v + 1) (u, v + 1) (1) """ if isinstance(P,(int,long,Integer)) and P == 0: R = PolynomialRing(self.value_ring(), 'x') return JacobianMorphism_divisor_class_field(self, (R(1),R(0))) elif isinstance(P,(list,tuple)): if len(P) == 1 and P[0] == 0: R = PolynomialRing(self.value_ring(), 'x') return JacobianMorphism_divisor_class_field(self, (R(1),R(0))) elif len(P) == 2: P1 = P[0]; P2 = P[1] if is_Polynomial(P1) and is_Polynomial(P2): return JacobianMorphism_divisor_class_field(self, tuple(P)) if is_SchemeMorphism(P1) and is_SchemeMorphism(P2): return self(P1) - self(P2) raise TypeError, "Argument P (= %s) must have length 2."%P elif isinstance(P,JacobianMorphism_divisor_class_field) and self == P.parent(): return P elif is_SchemeMorphism(P): x0 = P[0]; y0 = P[1] R, x = PolynomialRing(self.value_ring(), 'x').objgen() return self((x-x0,R(y0))) raise TypeError, "Argument P (= %s) does not determine a divisor class"%P
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sgi = set(range(fan.nrays()))
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sgi = set(range(fan.ngenerating_cones()))
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def star_generator_indices(self): r""" Return indices of generating cones of the "ambient fan" containing ``self``.
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K = NumberField_quadratic(polynomial, name, check, embedding, latex_name=latex_name)
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K = NumberField_quadratic(polynomial, name, latex_name, check, embedding)
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def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over `\QQ`. INPUT: - ``polynomial`` - a polynomial over `\QQ` or a number field, or a list of polynomials. - ``name`` - a string (default: 'a'), the name of the generator - ``check`` - bool (default: True); do type checking and irreducibility checking. - ``embedding`` - image of the generator in an ambient field (default: None) EXAMPLES:: sage: z = QQ['z'].0 sage: K = NumberField(z^2 - 2,'s'); K Number Field in s with defining polynomial z^2 - 2 sage: s = K.0; s s sage: s*s 2 sage: s^2 2 Constructing a relative number field:: sage: K.<a> = NumberField(x^2 - 2) sage: R.<t> = K[] sage: L.<b> = K.extension(t^3+t+a); L Number Field in b with defining polynomial t^3 + t + a over its base field sage: L.absolute_field('c') Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2 sage: a*b a*b sage: L(a) a sage: L.lift_to_base(b^3 + b) -a Constructing another number field:: sage: k.<i> = NumberField(x^2 + 1) sage: R.<z> = k[] sage: m.<j> = NumberField(z^3 + i*z + 3) sage: m Number Field in j with defining polynomial z^3 + i*z + 3 over its base field Number fields are globally unique:: sage: K.<a>= NumberField(x^3-5) sage: a^3 5 sage: L.<a>= NumberField(x^3-5) sage: K is L True Having different defining polynomials makes the fields different:: sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y') sage: k.<a> = NumberField(x^2 + 3) sage: m.<a> = NumberField(y^2 + 3) sage: k Number Field in a with defining polynomial x^2 + 3 sage: m Number Field in a with defining polynomial y^2 + 3 One can also define number fields with specified embeddings, may be used for arithmetic and deduce relations with other number fields which would not be valid for an abstract number field:: sage: K.<a> = NumberField(x^3-2, embedding=1.2) sage: RR.coerce_map_from(K) Composite map: From: Number Field in a with defining polynomial x^3 - 2 To: Real Field with 53 bits of precision Defn: Generic morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Real Lazy Field Defn: a -> 1.259921049894873? then Conversion via _mpfr_ method map: From: Real Lazy Field To: Real Field with 53 bits of precision sage: RR(a) 1.25992104989487 sage: 1.1 + a 2.35992104989487 sage: b = 1/(a+1); b 1/3*a^2 - 1/3*a + 1/3 sage: RR(b) 0.442493334024442 sage: L.<b> = NumberField(x^6-2, embedding=1.1) sage: L(a) b^2 sage: a + b b^2 + b Note that the image only needs to be specified to enough precision to distinguish roots, and is exactly computed to any needed precision:: sage: RealField(200)(a) 1.2599210498948731647672106072782283505702514647015079800820 One can embed into any other field:: sage: K.<a> = NumberField(x^3-2, embedding=CC.gen()-0.6) sage: CC(a) -0.629960524947436 + 1.09112363597172*I sage: L = Qp(5) sage: f = polygen(L)^3 - 2 sage: K.<a> = NumberField(x^3-2, embedding=f.roots()[0][0]) sage: a + L(1) 4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20) sage: L.<b> = NumberField(x^6-x^2+1/10, embedding=1) sage: K.<a> = NumberField(x^3-x+1/10, embedding=b^2) sage: a+b b^2 + b sage: CC(a) == CC(b)^2 True sage: K.coerce_embedding() Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Number Field in b with defining polynomial x^6 - x^2 + 1/10 Defn: a -> b^2 The ``QuadraticField`` and ``CyclotomicField`` constructors create an embedding by default unless otherwise specified. :: sage: K.<zeta> = CyclotomicField(15) sage: CC(zeta) 0.913545457642601 + 0.406736643075800*I sage: L.<sqrtn3> = QuadraticField(-3) sage: K(sqrtn3) 2*zeta^5 + 1 sage: sqrtn3 + zeta 2*zeta^5 + zeta + 1 An example involving a variable name that defines a function in PARI:: sage: theta = polygen(QQ, 'theta') sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M Number Field in z0 with defining polynomial theta^3 + 4 over its base field TESTS:: sage: x = QQ['x'].gen() sage: y = ZZ['y'].gen() sage: K = NumberField(x^3 + x + 3, 'a'); K Number Field in a with defining polynomial x^3 + x + 3 sage: K.defining_polynomial().parent() Univariate Polynomial Ring in x over Rational Field :: sage: L = NumberField(y^3 + y + 3, 'a'); L Number Field in a with defining polynomial y^3 + y + 3 sage: L.defining_polynomial().parent() Univariate Polynomial Ring in y over Rational Field :: sage: sage.rings.number_field.number_field._nf_cache = {} sage: K.<x> = CyclotomicField(5)[] sage: W.<a> = NumberField(x^2 + 1); W Number Field in a with defining polynomial x^2 + 1 over its base field sage: sage.rings.number_field.number_field._nf_cache = {} sage: W1 = NumberField(x^2+1,'a') sage: K.<x> = CyclotomicField(5)[] sage: W.<a> = NumberField(x^2 + 1); W Number Field in a with defining polynomial x^2 + 1 over its base field """ if name is None and names is None: raise TypeError, "You must specify the name of the generator." if not names is None: name = names if isinstance(polynomial, (list, tuple)): return NumberFieldTower(polynomial, name) name = sage.structure.parent_gens.normalize_names(1, name) if not isinstance(polynomial, polynomial_element.Polynomial): try: polynomial = polynomial.polynomial(QQ) except (AttributeError, TypeError): raise TypeError, "polynomial (=%s) must be a polynomial."%repr(polynomial) # convert ZZ to QQ R = polynomial.base_ring() Q = polynomial.parent().base_extend(R.fraction_field()) polynomial = Q(polynomial) if cache: key = (polynomial, polynomial.base_ring(), name, embedding, embedding.parent() if embedding is not None else None) if _nf_cache.has_key(key): K = _nf_cache[key]() if not K is None: return K if isinstance(R, NumberField_generic): S = R.extension(polynomial, name, check=check) if cache: _nf_cache[key] = weakref.ref(S) return S if polynomial.degree() == 2: K = NumberField_quadratic(polynomial, name, check, embedding, latex_name=latex_name) else: K = NumberField_absolute(polynomial, name, None, check, embedding, latex_name=latex_name) if cache: _nf_cache[key] = weakref.ref(K) return K
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K = NumberField_absolute(polynomial, name, None, check, embedding, latex_name=latex_name)
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K = NumberField_absolute(polynomial, name, latex_name, check, embedding)
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def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over `\QQ`. INPUT: - ``polynomial`` - a polynomial over `\QQ` or a number field, or a list of polynomials. - ``name`` - a string (default: 'a'), the name of the generator - ``check`` - bool (default: True); do type checking and irreducibility checking. - ``embedding`` - image of the generator in an ambient field (default: None) EXAMPLES:: sage: z = QQ['z'].0 sage: K = NumberField(z^2 - 2,'s'); K Number Field in s with defining polynomial z^2 - 2 sage: s = K.0; s s sage: s*s 2 sage: s^2 2 Constructing a relative number field:: sage: K.<a> = NumberField(x^2 - 2) sage: R.<t> = K[] sage: L.<b> = K.extension(t^3+t+a); L Number Field in b with defining polynomial t^3 + t + a over its base field sage: L.absolute_field('c') Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2 sage: a*b a*b sage: L(a) a sage: L.lift_to_base(b^3 + b) -a Constructing another number field:: sage: k.<i> = NumberField(x^2 + 1) sage: R.<z> = k[] sage: m.<j> = NumberField(z^3 + i*z + 3) sage: m Number Field in j with defining polynomial z^3 + i*z + 3 over its base field Number fields are globally unique:: sage: K.<a>= NumberField(x^3-5) sage: a^3 5 sage: L.<a>= NumberField(x^3-5) sage: K is L True Having different defining polynomials makes the fields different:: sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y') sage: k.<a> = NumberField(x^2 + 3) sage: m.<a> = NumberField(y^2 + 3) sage: k Number Field in a with defining polynomial x^2 + 3 sage: m Number Field in a with defining polynomial y^2 + 3 One can also define number fields with specified embeddings, may be used for arithmetic and deduce relations with other number fields which would not be valid for an abstract number field:: sage: K.<a> = NumberField(x^3-2, embedding=1.2) sage: RR.coerce_map_from(K) Composite map: From: Number Field in a with defining polynomial x^3 - 2 To: Real Field with 53 bits of precision Defn: Generic morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Real Lazy Field Defn: a -> 1.259921049894873? then Conversion via _mpfr_ method map: From: Real Lazy Field To: Real Field with 53 bits of precision sage: RR(a) 1.25992104989487 sage: 1.1 + a 2.35992104989487 sage: b = 1/(a+1); b 1/3*a^2 - 1/3*a + 1/3 sage: RR(b) 0.442493334024442 sage: L.<b> = NumberField(x^6-2, embedding=1.1) sage: L(a) b^2 sage: a + b b^2 + b Note that the image only needs to be specified to enough precision to distinguish roots, and is exactly computed to any needed precision:: sage: RealField(200)(a) 1.2599210498948731647672106072782283505702514647015079800820 One can embed into any other field:: sage: K.<a> = NumberField(x^3-2, embedding=CC.gen()-0.6) sage: CC(a) -0.629960524947436 + 1.09112363597172*I sage: L = Qp(5) sage: f = polygen(L)^3 - 2 sage: K.<a> = NumberField(x^3-2, embedding=f.roots()[0][0]) sage: a + L(1) 4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20) sage: L.<b> = NumberField(x^6-x^2+1/10, embedding=1) sage: K.<a> = NumberField(x^3-x+1/10, embedding=b^2) sage: a+b b^2 + b sage: CC(a) == CC(b)^2 True sage: K.coerce_embedding() Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Number Field in b with defining polynomial x^6 - x^2 + 1/10 Defn: a -> b^2 The ``QuadraticField`` and ``CyclotomicField`` constructors create an embedding by default unless otherwise specified. :: sage: K.<zeta> = CyclotomicField(15) sage: CC(zeta) 0.913545457642601 + 0.406736643075800*I sage: L.<sqrtn3> = QuadraticField(-3) sage: K(sqrtn3) 2*zeta^5 + 1 sage: sqrtn3 + zeta 2*zeta^5 + zeta + 1 An example involving a variable name that defines a function in PARI:: sage: theta = polygen(QQ, 'theta') sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M Number Field in z0 with defining polynomial theta^3 + 4 over its base field TESTS:: sage: x = QQ['x'].gen() sage: y = ZZ['y'].gen() sage: K = NumberField(x^3 + x + 3, 'a'); K Number Field in a with defining polynomial x^3 + x + 3 sage: K.defining_polynomial().parent() Univariate Polynomial Ring in x over Rational Field :: sage: L = NumberField(y^3 + y + 3, 'a'); L Number Field in a with defining polynomial y^3 + y + 3 sage: L.defining_polynomial().parent() Univariate Polynomial Ring in y over Rational Field :: sage: sage.rings.number_field.number_field._nf_cache = {} sage: K.<x> = CyclotomicField(5)[] sage: W.<a> = NumberField(x^2 + 1); W Number Field in a with defining polynomial x^2 + 1 over its base field sage: sage.rings.number_field.number_field._nf_cache = {} sage: W1 = NumberField(x^2+1,'a') sage: K.<x> = CyclotomicField(5)[] sage: W.<a> = NumberField(x^2 + 1); W Number Field in a with defining polynomial x^2 + 1 over its base field """ if name is None and names is None: raise TypeError, "You must specify the name of the generator." if not names is None: name = names if isinstance(polynomial, (list, tuple)): return NumberFieldTower(polynomial, name) name = sage.structure.parent_gens.normalize_names(1, name) if not isinstance(polynomial, polynomial_element.Polynomial): try: polynomial = polynomial.polynomial(QQ) except (AttributeError, TypeError): raise TypeError, "polynomial (=%s) must be a polynomial."%repr(polynomial) # convert ZZ to QQ R = polynomial.base_ring() Q = polynomial.parent().base_extend(R.fraction_field()) polynomial = Q(polynomial) if cache: key = (polynomial, polynomial.base_ring(), name, embedding, embedding.parent() if embedding is not None else None) if _nf_cache.has_key(key): K = _nf_cache[key]() if not K is None: return K if isinstance(R, NumberField_generic): S = R.extension(polynomial, name, check=check) if cache: _nf_cache[key] = weakref.ref(S) return S if polynomial.degree() == 2: K = NumberField_quadratic(polynomial, name, check, embedding, latex_name=latex_name) else: K = NumberField_absolute(polynomial, name, None, check, embedding, latex_name=latex_name) if cache: _nf_cache[key] = weakref.ref(K) return K
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def __init__(self, polynomial, name=None, check=True, embedding=None, latex_name=None):
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def __init__(self, polynomial, name=None, latex_name=None, check=True, embedding=None):
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def __init__(self, polynomial, name=None, check=True, embedding=None, latex_name=None): """ Create a quadratic number field. EXAMPLES:: sage: k.<a> = QuadraticField(5, check=False); k Number Field in a with defining polynomial x^2 - 5 Don't do this:: sage: k.<a> = QuadraticField(4, check=False); k Number Field in a with defining polynomial x^2 - 4
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Initializes base class Ellipse.
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Initializes base class ``Ellipse``.
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def __init__(self, x, y, r1, r2, angle, options): """ Initializes base class Ellipse.
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The bounding box is computed as minimal as possible.
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The bounding box is computed to be as minimal as possible.
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def get_minmax_data(self): """ Returns a dictionary with the bounding box data.
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An example without angle::
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An example without an angle::
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def get_minmax_data(self): """ Returns a dictionary with the bounding box data.
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The same example with a rotation of angle pi/2::
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The same example with a rotation of angle `\pi/2`::
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def get_minmax_data(self): """ Returns a dictionary with the bounding box data.
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Return the allowed options for the Ellipse class.
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Return the allowed options for the ``Ellipse`` class.
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def _allowed_options(self): """ Return the allowed options for the Ellipse class.
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String representation of Ellipse primitive.
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String representation of ``Ellipse`` primitive.
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def _repr_(self): """ String representation of Ellipse primitive.
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Plot 3d is not implemented.
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Plotting in 3D is not implemented.
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def plot3d(self): r""" Plot 3d is not implemented.
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""" return Vobj.evaluated_on(self)
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If you pass a vector, it is assumed to be the coordinate vector of a point:: sage: ineq.eval( vector(ZZ, [3,2]) ) -4 """ try: return Vobj.evaluated_on(self) except AttributeError: return self.A() * Vobj + self.b()
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def eval(self, Vobj): r""" Evaluates the left hand side `A\vec{x}+b` on the given vertex/ray/line. NOTES: * Evaluating on a vertex returns `A\vec{x}+b` * Evaluating on a ray returns `A\vec{r}`. Only the sign or whether it is zero is meaningful. * Evaluating on a line returns `A\vec{l}`. Only whether it is zero or not is meaningful.
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is_inequality.__doc__ = Hrepresentation.is_inequality.__doc__
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def is_inequality(self): """ Returns True since this is, by construction, an inequality.
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"""
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If you pass a vector, it is assumed to be the coordinate vector of a point:: sage: P = Polyhedron(vertices=[[1,1],[1,-1],[-1,1],[-1,-1]]) sage: p = vector(ZZ, [1,0] ) sage: [ ieq.interior_contains(p) for ieq in P.inequality_generator() ] [True, True, True, False] """ try: if Vobj.is_vector(): return self.polyhedron()._is_positive( self.eval(Vobj) ) except AttributeError: pass
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def interior_contains(self, Vobj): """ Tests whether the interior of the halfspace (excluding its boundary) defined by the inequality contains the given vertex/ray/line.
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is_equation.__doc__ = Hrepresentation.is_equation.__doc__
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def is_equation(self): """ Tests if this object is an equation. By construction, it must be.
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is_vertex.__doc__ = Vrepresentation.is_vertex.__doc__
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def is_vertex(self): """ Tests if this object is a vertex. By construction it always is.
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is_ray.__doc__ = Vrepresentation.is_ray.__doc__
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def is_ray(self): """ Tests if this object is a ray. Always True by construction.
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is_line.__doc__ = Vrepresentation.is_line.__doc__
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def is_line(self): """ Tests if the object is a line. By construction it must be.
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Returns the identity projection.
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Returns the identity projection of the polyhedron.
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def identity(self): """ Returns the identity projection.
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identity.__doc__ = projection_func_identity.__doc__
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def identity(self): """ Returns the identity projection.
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(x(t),y(t)) such that y(t)^2 = f(x(t)) and t
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(x(t),y(t)) such that y(t)^2 = f(x(t)), where t
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def local_coord(self, P, prec = 20, name = 't'): """ If P is not infinity, calls the appropriate local_coordinates function.
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sage: browse_sage_doc(identity_matrix, 'rst')[-60:-5] 'MatrixSpace of 3 by 3 sparse matrices over Integer Ring'
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sage: browse_sage_doc(identity_matrix, 'rst')[-107:-47] 'Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring'
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'def identity_matrix'
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Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the matching polynomial of G is equal to that of G' minus that of G''.
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Computes the matching polynomial of the graph `G`. If `p(G, k)` denotes the number of `k`-matchings (matchings with `k` edges) in `G`, then the matching polynomial is defined as [Godsil93]_: .. MATH:: \mu(x)=\sum_{k \geq 0} (-1)^k p(G,k) x^{n-2k}
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the matching polynomial of G is equal to that of G' minus that of G''.
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- ``complement`` - (default: True) whether to use a simple formula to compute the matching polynomial from that of the graphs complement
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- ``complement`` - (default: ``True``) whether to use Godsil's duality theorem to compute the matching polynomial from that of the graphs complement (see ALGORITHM).
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the matching polynomial of G is equal to that of G' minus that of G''.
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NOTE: The ``complement`` option uses matching polynomials of complete graphs, which are cached. So if you are crazy enough to try computing the matching polynomial on a graph with millions of vertices, you might not want to use this option, since it will end up caching millions of polynomials of degree in the millions.
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the matching polynomial of G is equal to that of G' minus that of G''.
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assert z.denominator() == 1, "bug in global_integral_model: %s" % ai
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assert z.is_integral(), "bug in global_integral_model: %s" % list(ai)
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def global_integral_model(self): r""" Return a model of self which is integral at all primes. EXAMPLES::
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For infinite periodic words (resp. for finite words of type `u^i u[0:j]`), the reduced Rauzy graph of order `n` (resp. for `n` smaller or equal to `(i-1)|u|+j`) is the directed graph whose unique vertex is the prefix `p` of length `n` of self and which has an only edge which is a loop on `p` labelled by `w[n+1:|w|] p` where `w` is the unique return word to `p`. In other cases, it is the directed graph defined as followed. Let `G_n` be the Rauzy graph of order `n` of self. The vertices are the vertices of `G_n` that are either special or not prolongable to the right of to the left. For each couple (`u`, `v`) of such vertices and each directed path in `G_n` from `u` to `v` that contains no other vertices that are special, there is an edge from `u` to `v` in the reduced Rauzy graph of order `n` whose label is the label of the path in `G_n`. NOTE: In the case of infinite recurrent non periodic words, this definition correspond to the following one that can be found in [1] and [2] where a simple path is a path that begins with a special factor, ends with a special factor and contains no other vertices that are special: The reduced Rauzy graph of factors of length `n` is obtained from `G_n` by replacing each simple path `P=v_1 v_2 ... v_{\ell}` with an edge `v_1 v_{\ell}` whose label is the concatenation of the labels of the edges of `P`. INPUT: - ``n`` - integer OUTPUT: Digraph
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For infinite periodic words (resp. for finite words of type `u^i u[0:j]`), the reduced Rauzy graph of order `n` (resp. for `n` smaller or equal to `(i-1)|u|+j`) is the directed graph whose unique vertex is the prefix `p` of length `n` of self and which has an only edge which is a loop on `p` labelled by `w[n+1:|w|] p` where `w` is the unique return word to `p`. In other cases, it is the directed graph defined as followed. Let `G_n` be the Rauzy graph of order `n` of self. The vertices are the vertices of `G_n` that are either special or not prolongable to the right or to the left. For each couple (`u`, `v`) of such vertices and each directed path in `G_n` from `u` to `v` that contains no other vertices that are special, there is an edge from `u` to `v` in the reduced Rauzy graph of order `n` whose label is the label of the path in `G_n`. .. NOTE:: In the case of infinite recurrent non periodic words, this definition correspond to the following one that can be found in [1] and [2] where a simple path is a path that begins with a special factor, ends with a special factor and contains no other vertices that are special: The reduced Rauzy graph of factors of length `n` is obtained from `G_n` by replacing each simple path `P=v_1 v_2 ... v_{\ell}` with an edge `v_1 v_{\ell}` whose label is the concatenation of the labels of the edges of `P`.
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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:: For the Fibonacci word: ::
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For the Fibonacci word::
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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:: For periodic words: ::
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For periodic words::
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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:: For ultimataly periodic words: ::
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For ultimately periodic words::
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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return words," Advances in Applied Mathematics 42, no. 1 60-74
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return words," Advances in Applied Mathematics 42 (2009) 60-74.
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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155,(3-4) : 251-263, 2008.
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155 (2008) 251-263.
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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g.add_edge(i,o,g.edge_label(i,v)[0]+g.edge_label(v,o)[0])
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g.add_edge(i,o,g.edge_label(i,v)[0]*g.edge_label(v,o)[0])
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def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self.
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def order(self, *gens, **kwds):
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def order(self, *args, **kwds):
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def order(self, *gens, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: True), whether to check that each generator is integral. - ``check_rank`` - bool (default: True), whether to check that the ring generated by gens is of full rank. - ``allow_subfield`` - bool (default: False), if True and the generators do not generate an order, i.e., they generate a subring of smaller rank, instead of raising an error, return an order in a smaller number field. EXAMPLES:: sage: k.<i> = NumberField(x^2 + 1) sage: k.order(2*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(10*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(3) Traceback (most recent call last): ... ValueError: the rank of the span of gens is wrong sage: k.order(i/2) Traceback (most recent call last): ... ValueError: each generator must be integral Alternatively, an order can be constructed by adjoining elements to `\ZZ`: """ if len(gens) == 0: return NumberField_generic.order(self) if len(gens) == 1 and isinstance(gens[0], (list, tuple)): gens = gens[0] gens = map(self, gens) import sage.rings.number_field.order as order return order.absolute_order_from_ring_generators(gens, **kwds)
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"""
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sage: K.<a> = NumberField(x^3 - 2) sage: ZZ[a] Order in Number Field in a0 with defining polynomial x^3 - 2 TESTS: We verify that trac sage: K.<a> = NumberField(x^4 + 4*x^2 + 2) sage: B = K.integral_basis() sage: K.order(*B) Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2 sage: K.order(B) Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2 sage: K.order(gens=B) Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2 """ gens = kwds.pop('gens', args)
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def order(self, *gens, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: True), whether to check that each generator is integral. - ``check_rank`` - bool (default: True), whether to check that the ring generated by gens is of full rank. - ``allow_subfield`` - bool (default: False), if True and the generators do not generate an order, i.e., they generate a subring of smaller rank, instead of raising an error, return an order in a smaller number field. EXAMPLES:: sage: k.<i> = NumberField(x^2 + 1) sage: k.order(2*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(10*i) Order in Number Field in i with defining polynomial x^2 + 1 sage: k.order(3) Traceback (most recent call last): ... ValueError: the rank of the span of gens is wrong sage: k.order(i/2) Traceback (most recent call last): ... ValueError: each generator must be integral Alternatively, an order can be constructed by adjoining elements to `\ZZ`: """ if len(gens) == 0: return NumberField_generic.order(self) if len(gens) == 1 and isinstance(gens[0], (list, tuple)): gens = gens[0] gens = map(self, gens) import sage.rings.number_field.order as order return order.absolute_order_from_ring_generators(gens, **kwds)
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Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points.
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Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. Used internally by the :meth:`~rank`, :meth:`~rank_bounds` and :meth:`~gens` methods.
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def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. INPUT:
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Uses Denis Simon's GP/PARI scripts from \url{http://www.math.unicaen.fr/~simon/}.
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Uses Denis Simon's GP/PARI scripts from http://www.math.unicaen.fr/~simon/.
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def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. INPUT:
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Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached.
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Returns the lower and upper bounds using :meth:`~simon_two_descent`. The results of :meth:`~simon_two_descent` are cached.
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def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached.
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These optional parameters control the Simon two descent algorithm.
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The optional parameters control the Simon two descent algorithm; see the documentation of :meth:`~simon_two_descent` for more details.
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def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached.
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\url{http://www.math.unicaen.fr/~simon/}.
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http://www.math.unicaen.fr/~simon/.
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def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached.
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return this. Otherwise, we return the upper and lower bounds with a warning that these are not the same.
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return this. Otherwise, we raise a ValueError with an error message specifying the upper and lower bounds.
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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Note: For non-quadratic number fields, this code does return, but it takes a long time.
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For non-quadratic number fields, this code does return, but it takes a long time.
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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Here is a curve with two-torsion, so here the algorithm gives bounds on the rank::
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Here is a curve with two-torsion, so here the bounds given by the algorithm do not uniquely determine the rank::
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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ValueError: There is insufficient data to determine the rank.
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ValueError: There is insufficient data to determine the rank - 2-descent gave lower bound 1 and upper bound 2
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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\url{http://www.math.unicaen.fr/~simon/}.
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http://www.math.unicaen.fr/~simon/.
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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raise ValueError, 'There is insufficient data to determine the rank.'
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raise ValueError, 'There is insufficient data to determine the rank - 2-descent gave lower bound %s and upper bound %s' % (lower, upper)
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def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined.
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Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators.
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Returns some generators of this elliptic curve. Check :meth:`~rank` or :meth:`~rank_bounds` to verify the number of generators. .. NOTE:: The optional parameters control the Simon two descent algorithm; see the documentation of :meth:`~simon_two_descent` for more details.
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def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators.
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Note: For non-quadratic number fields, this code does return, but it takes a long time.
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For non-quadratic number fields, this code does return, but it takes a long time.
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def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators.
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Here is a curve with two-torsion, so here the algorithm gives bounds on the rank::
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Here is a curve with two-torsion, so here the algorithm does not uniquely determine the rank::
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def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators.
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\url{http://www.math.unicaen.fr/~simon/}.
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http://www.math.unicaen.fr/~simon/.
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def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators.
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SL(n); you also some information about representations of E6
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SL(n); you also lose some information about representations of E6
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def WeylCharacterRing(ct, base_ring=ZZ, prefix=None, cache=False, style="lattice"): r""" A class for rings of Weyl characters. The Weyl character is a character of a semisimple (or reductive) Lie group or algebra. They form a ring, in which the addition and multiplication correspond to direct sum and tensor product of representations. INPUT: - ``ct`` -- The Cartan Type OPTIONAL ARGUMENTS: - ``base_ring`` -- (default: `\ZZ`) - ``prefix`` -- (default: an automatically generated prefix based on Cartan type) - ``cache`` -- (default False) setting cache = True is a substantial speedup at the expense of some memory. - ``style`` -- (default "lattice") can be set style = "coroots" to obtain an alternative representation of the elements. If no prefix specified, one is generated based on the Cartan type. It is good to name the ring after the prefix, since then it can parse its own output. EXAMPLES:: sage: G2 = WeylCharacterRing(['G',2]) sage: [fw1,fw2] = G2.fundamental_weights() sage: 2*G2(2*fw1+fw2) 2*G2(4,-1,-3) sage: 2*G2(4,-1,-3) 2*G2(4,-1,-3) sage: G2(4,-1,-3).degree() 189 Note that since the ring was named `G_2` after its default prefix, it was able to parse its own output. You do not have to use the default prefix. Thus: EXAMPLES:: sage: R = WeylCharacterRing(['B',3], prefix='R') sage: chi = R(R.fundamental_weights()[3]); chi R(1/2,1/2,1/2) sage: R(1/2,1/2,1/2) == chi True You may choose an alternative style of labeling the elements. If you create the ring with the option style="coroots" then the integers in the label are not the components of the highest weight vector, but rather the coefficients when the highest weight vector is decomposed into a product of irreducibles. These coefficients are the values of the coroots on the highest weight vector. In the coroot style the Lie group or Lie algebra is treated as semisimple, so you lose the distinction between GL(n) and SL(n); you also some information about representations of E6 and E7 for the same reason. The coroot style gives you output that is comparable to that in Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras (Marcel Dekker, 1981). EXAMPLES:: sage: B3 = WeylCharacterRing("B3",style="coroots") sage: [fw1,fw2,fw3]=B3.fundamental_weights() sage: fw1+fw3 (3/2, 1/2, 1/2) sage: B3(fw1+fw3) B3(1,0,1) sage: B3(1,0,1) B3(1,0,1) For type ['A',r], the coroot representation carries less information, since elements of the weight lattice that are orthogonal to the coroots are represented as zero. This means that with the default style you can represent the determinant, but not in the coroot style. In the coroot style, elements of the Weyl character ring represent characters of SL(r+1,CC), while in the default style, they represent characters of GL(r+1,CC). EXAMPLES:: sage: A2 = WeylCharacterRing("A2") sage: L = A2.space() sage: [A2(L.det()), A2(L(0))] [A2(1,1,1), A2(0,0,0)] sage: A2(L.det()) == A2(L(0)) False sage: A2 = WeylCharacterRing("A2", style="coroots") sage: [A2(L.det()), A2(L(0))] [A2(0,0), A2(0,0)] sage: A2(L.det()) == A2(L(0)) True The multiplication in a Weyl character ring corresponds to the product of characters, which you can use to determine the decomposition of tensor products into irreducibles. For example, let us compute the tensor product of the standard and spin representations of Spin(7). EXAMPLES:: sage: B3 = WeylCharacterRing("B3") sage: [fw1,fw2,fw3]=B3.fundamental_weights() sage: [B3(fw1).degree(),B3(fw3).degree()] [7, 8] sage: B3(fw1)*B3(fw3) B3(1/2,1/2,1/2) + B3(3/2,1/2,1/2) The name of the irreducible representation encodes the highest weight vector. TESTS:: sage: F4 = WeylCharacterRing(['F',4], cache = True) sage: [fw1,fw2,fw3,fw4] = F4.fundamental_weights() sage: chi = F4(fw4); chi, chi.degree() (F4(1,0,0,0), 26) sage: chi^2 F4(0,0,0,0) + F4(1,0,0,0) + F4(1,1,0,0) + F4(3/2,1/2,1/2,1/2) + F4(2,0,0,0) sage: [x.degree() for x in [F4(0,0,0,0), F4(1,0,0,0), F4(1,1,0,0), F4(3/2,1/2,1/2,1/2), F4(2,0,0,0)]] [1, 26, 52, 273, 324] You can produce a list of the irreducible elements of an irreducible character. EXAMPLES:: sage: R = WeylCharacterRing(['A',2], prefix = R) sage: sorted(R([2,1,0]).mlist()) [[(1, 1, 1), 2], [(1, 2, 0), 1], [(1, 0, 2), 1], [(2, 1, 0), 1], [(2, 0, 1), 1], [(0, 1, 2), 1], [(0, 2, 1), 1]] """ ct = cartan_type.CartanType(ct) return cache_wcr(ct, base_ring=base_ring, prefix=prefix, cache=cache, style=style)
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"automorphic", "symmetric", "extended", "triality" or "miscellaneous". The use of these rules will be explained next. After the examples we will explain how to write your own branching rules for cases that we have omitted.
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"automorphic", "symmetric", "extended", "orthogonal_sum", "tensor", "triality" or "miscellaneous". The use of these rules will be explained next. After the examples we will explain how to write your own branching rules for cases that we have omitted.
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def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Willenbring, Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601-1626, McKay and Patera, Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras (Marcel Dekker, 1981), and Fauser, Jarvis, King and Wybourne, New branching rules induced by plethysm. J. Phys. A 39 (2006), no. 11, 2611--2655. INPUT: - ``chi`` - a character of G - ``R`` - the Weyl Character Ring of G - ``S`` - the Weyl Character Ring of H - ``rule`` - a set of r dominant weights in H where r is the rank of G. You may use a predefined rule by specifying rule = one of"levi", "automorphic", "symmetric", "extended", "triality" or "miscellaneous". The use of these rules will be explained next. After the examples we will explain how to write your own branching rules for cases that we have omitted. To explain the predefined rules we survey the most important branching rules. These may be classified into several cases, and once this is understood, the detailed classification can be read off from the Dynkin diagrams. Dynkin classified the maximal subgroups of Lie groups in Mat. Sbornik N.S. 30(72):349-462 (1952). We will list give predefined rules that cover most cases where the branching rule is to a maximal subgroup. For convenience, we also give some branching rules to subgroups that are not maximal. For example, a Levi subgroup may or may not be maximal. LEVI TYPE. These can be read off from the Dynkin diagram. If removing a node from the Dynkin diagram produces another Dynkin diagram, there is a branching rule. Currently we require that the smaller diagram be connected. For these rules use the option rule="levi":: ['A',r] => ['A',r-1] ['B',r] => ['A',r-1] ['B',r] => ['B',r-1] ['C',r] => ['A',r-1] ['C',r] => ['C',r-1] ['D',r] => ['A',r-1] ['D',r] => ['D',r-1] ['E',r] => ['A',r-1] r = 7,8 ['E',r] => ['D',r-1] r = 6,7,8 ['E',r] => ['E',r-1] F4 => B3 F4 => C3 G2 => A1 (short root) Not all Levi subgroups are maximal subgroups. If the Levi is not maximal there may or may not be a preprogrammed rule="levi" for it. If there is not, the branching rule may still be obtained by going through an intermediate subgroup that is maximal using rule="extended". Thus the other Levi branching rule from G2 => A1 corresponding to the long root is available by first branching G2 => A_2 then A2 => A1. Similarly the branching rules to the Levi subgroup:: ['E',r] => ['A',r-1] r = 6,7,8 may be obtained by first branching E6=>A5xA1, E7=>A7 or E8=>A8. AUTOMORPHIC TYPE. If the Dynkin diagram has a symmetry, then there is an automorphism that is a special case of a branching rule. There is also an exotic "triality" automorphism of D4 having order 3. Use rule="automorphic" or (for D4) rule="triality":: ['A',r] => ['A',r] ['D',r] => ['D',r] E6 => E6 SYMMETRIC TYPE. Related to the automorphic type, when either the Dynkin diagram or the extended diagram has a symmetry there is a branching rule to the subalgebra (or subgroup) of invariants under the automorphism. Use rule="symmetric". The last branching rule, D4=>G2 is not to a maximal subgroup since D4=>B3=>G2, but it is included for convenience. :: ['A',2r+1] => ['B',r] ['A',2r] => ['C',r] ['A',2r] => ['D',r] ['D',r] => ['B',r-1] E6 => F4 D4 => G2 EXTENDED TYPE. If removing a node from the extended Dynkin diagram results in a Dynkin diagram, then there is a branching rule. Use rule="extended" for these. We will also use this classification for some rules that are not of this type, mainly involving type B, such as D6 => B3xB3. Here is the extended Dynkin diagram for D6:: 0 6 O O | | | | O---O---O---O---O 1 2 3 4 6 Removing the node 3 results in an embedding D3xD3 -> D6. This corresponds to the embedding SO(6)xSO(6) -> SO(12), and is of extended type. On the other hand the embedding SO(5)xSO(7)-->SO(12) (e.g. B2xB3 -> D6) cannot be explained this way but for uniformity is implemented under rule="extended". Using rule="extended" you can get any branching rule SO(n) => SO(a) x SO(b) x SO(c) x ... where n = a+b+c+ ... Sp(2n) => Sp(2a) x Sp(2b) x Sp(2c) x ... where n = a+b+c+ ... where O(a) = ['D',r] (a=2r) or ['B',r] (a=2r+1) and Sp(2r)=['C',r]. The following rules are implemented as special cases of rule="extended". :: E6 => A5xA1, A2xA2xA2 E7 => A7, D6xA1, A3xA3xA1 E8 => A8, D8, E7xA1, A4xA4, D5xA3, E6xA2 F4 => B4, C3xA1, A2xA2, A3xA1 G2 => A1xA1 Note that E8 has only a limited number of representations of reasonably low degree. TENSOR: There are branching rules: :: ['A', rs-1] => ['A',r-1] x ['A',s-1] ['B',2rs+r+s] => ['B',r] x ['B',s] ['D',2rs+s] => ['B',r] x ['D',s] ['D',2rs] => ['D',r] x ['D',s] ['D',2rs] => ['C',r] x ['C',s] ['C',2rs+s] => ['B',r] x ['C',s] ['C',2rs] => ['C',r] x ['D',s]. corresponding to the tensor product homomorphism. For type A, the homomorphism is GL(r) x GL(s) -> GL(rs). For the classical types, the relevant fact is that if V,W are orthogonal or symplectic spaces, that is, spaces endowed with symmetric or skew-symmetric bilinear forms, then V tensor W is also an orthogonal space (if V and W are both orthogonal or both symplectic) or symplectic (if one of V and W is orthogonal and the other symplectic). The corresponding branching rules are obtained using rule="tensor". SYMMETRIC POWER: The k-th symmetric and exterior power homomorphisms map GL(n) --> GL(binomial(n+k-1,k)) and GL(binomial(n,k)). The corresponding branching rules are not implemented but a special case is. The k-th symmetric power homomorphism SL(2) --> GL(k+1) has its image inside of SO(2r+1) if k=2r and inside of Sp(2r) if k=2r-1. Hence there are branching rules:: ['B',r] => A1 ['C',r] => A1 and these may be obtained using the rule "symmetric_power". MISCELLANEOUS: Use rule="miscellaneous" for the following rules:: B3 => G2 F4 => G2xA1 (not implemented yet) BRANCHING RULES FROM PLETHYSMS Nearly all branching rules G => H where G is of type A,B,C or D are covered by the preceding rules. The function branching_rules_from_plethysm covers the remaining cases. ISOMORPHIC TYPE: Although not usually referred to as a branching rule, the effects of the accidental isomorphisms may be handled using rule="isomorphic":: B2 => C2 C2 => B2 A3 => D3 D3 => A3 D2 => A1xA1 B1 => A1 C1 => A1 EXAMPLES: (Levi type) :: sage: A1 = WeylCharacterRing("A1") sage: A2 = WeylCharacterRing("A2") sage: A3 = WeylCharacterRing("A3") sage: A4 = WeylCharacterRing("A4") sage: A5 = WeylCharacterRing("A5") sage: B2 = WeylCharacterRing("B2") sage: B3 = WeylCharacterRing("B3") sage: B4 = WeylCharacterRing("B4") sage: C2 = WeylCharacterRing("C2") sage: C3 = WeylCharacterRing("C3") sage: D3 = WeylCharacterRing("D3") sage: D4 = WeylCharacterRing("D4") sage: D5 = WeylCharacterRing("D5") sage: G2 = WeylCharacterRing("G2") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: [B3(w).branch(A2,rule="levi") for w in B3.fundamental_weights()] [A2(0,0,-1) + A2(0,0,0) + A2(1,0,0), A2(0,-1,-1) + A2(0,0,-1) + A2(0,0,0) + A2(1,0,-1) + A2(1,0,0) + A2(1,1,0), A2(-1/2,-1/2,-1/2) + A2(1/2,-1/2,-1/2) + A2(1/2,1/2,-1/2) + A2(1/2,1/2,1/2)] The last example must be understood as follows. The representation of B3 being branched is spin, which is not a representation of SO(7) but of its double cover spin(7). The group A2 is really GL(3) and the double cover of SO(7) induces a cover of GL(3) that is trivial over SL(3) but not over the center of GL(3). The weight lattice for this GL(3) consists of triples (a,b,c) of half integers such that a-b and b-c are in `\ZZ`, and this is reflected in the last decomposition. :: sage: [C3(w).branch(A2,rule="levi") for w in C3.fundamental_weights()] [A2(0,0,-1) + A2(1,0,0), A2(0,-1,-1) + A2(1,0,-1) + A2(1,1,0), A2(-1,-1,-1) + A2(1,-1,-1) + A2(1,1,-1) + A2(1,1,1)] sage: [D4(w).branch(A3,rule="levi") for w in D4.fundamental_weights()] [A3(0,0,0,-1) + A3(1,0,0,0), A3(0,0,-1,-1) + A3(0,0,0,0) + A3(1,0,0,-1) + A3(1,1,0,0), A3(1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,-1/2), A3(-1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,1/2)] sage: [B3(w).branch(B2,rule="levi") for w in B3.fundamental_weights()] [2*B2(0,0) + B2(1,0), B2(0,0) + 2*B2(1,0) + B2(1,1), 2*B2(1/2,1/2)] sage: C3 = WeylCharacterRing(['C',3]) sage: [C3(w).branch(C2,rule="levi") for w in C3.fundamental_weights()] [2*C2(0,0) + C2(1,0), C2(0,0) + 2*C2(1,0) + C2(1,1), C2(1,0) + 2*C2(1,1)] sage: [D5(w).branch(D4,rule="levi") for w in D5.fundamental_weights()] [2*D4(0,0,0,0) + D4(1,0,0,0), D4(0,0,0,0) + 2*D4(1,0,0,0) + D4(1,1,0,0), D4(1,0,0,0) + 2*D4(1,1,0,0) + D4(1,1,1,0), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2)] sage: G2(1,0,-1).branch(A1,rule="levi") A1(0,-1) + A1(1,-1) + A1(1,0) sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: fw = E6.fundamental_weights() sage: [E6(fw[i]).branch(D5,rule="levi") for i in [1,2,6]] # long time (3s) [D5(0,0,0,0,0) + D5(0,0,0,0,1) + D5(1,0,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(0,0,0,0,1) + D5(0,1,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(1,0,0,0,0)] sage: E7=WeylCharacterRing("E7",style="coroots") sage: D6=WeylCharacterRing("D6",style="coroots") sage: fw = E7.fundamental_weights() sage: [E7(fw[i]).branch(D6,rule="levi") for i in [1,2,7]] # long time (26s) [3*D6(0,0,0,0,0,0) + 2*D6(0,0,0,0,1,0) + D6(0,1,0,0,0,0), 3*D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0) + 2*D6(0,0,1,0,0,0) + D6(1,0,0,0,1,0), D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0)] sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8=WeylCharacterRing("E8",style="coroots",cache=True) sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8(1,0,0,0,0,0,0,0).branch(D7,rule="levi") # not tested (very long time) (160s) 3*D7(0,0,0,0,0,0,0) + 2*D7(0,0,0,0,0,1,0) + 2*D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) + 2*D7(0,0,1,0,0,0,0) + D7(0,0,0,1,0,0,0) + D7(1,0,0,0,0,1,0) + D7(1,0,0,0,0,0,1) + D7(2,0,0,0,0,0,0) sage: E8(0,0,0,0,0,0,0,1).branch(D7,rule="levi") # long time (3s) D7(0,0,0,0,0,0,0) + D7(0,0,0,0,0,1,0) + D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) sage: [F4(fw).branch(B3,rule="levi") for fw in F4.fundamental_weights()] # long time (36s) [B3(0,0,0) + 2*B3(1/2,1/2,1/2) + 2*B3(1,0,0) + B3(1,1,0), B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 5*B3(1,0,0) + 7*B3(1,1,0) + 3*B3(1,1,1) + 6*B3(3/2,1/2,1/2) + 2*B3(3/2,3/2,1/2) + B3(2,0,0) + 2*B3(2,1,0) + B3(2,1,1), 3*B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 4*B3(1,0,0) + 3*B3(1,1,0) + B3(1,1,1) + 2*B3(3/2,1/2,1/2), 3*B3(0,0,0) + 2*B3(1/2,1/2,1/2) + B3(1,0,0)] sage: [F4(fw).branch(C3,rule="levi") for fw in F4.fundamental_weights()] # long time (6s) [3*C3(0,0,0) + 2*C3(1,1,1) + C3(2,0,0), 3*C3(0,0,0) + 6*C3(1,1,1) + 4*C3(2,0,0) + 2*C3(2,1,0) + 3*C3(2,2,0) + C3(2,2,2) + C3(3,1,0) + 2*C3(3,1,1), 2*C3(1,0,0) + 3*C3(1,1,0) + C3(2,0,0) + 2*C3(2,1,0) + C3(2,1,1), 2*C3(1,0,0) + C3(1,1,0)] sage: A1xA1 = WeylCharacterRing("A1xA1") sage: [A3(hwv).branch(A1xA1,rule="levi") for hwv in A3.fundamental_weights()] [A1xA1(0,0,1,0) + A1xA1(1,0,0,0), A1xA1(0,0,1,1) + A1xA1(1,0,1,0) + A1xA1(1,1,0,0), A1xA1(1,0,1,1) + A1xA1(1,1,1,0)] sage: A1xB1=WeylCharacterRing("A1xB1",style="coroots") sage: [B3(x).branch(A1xB1,rule="levi") for x in B3.fundamental_weights()] [A1xB1(0,2) + 2*A1xB1(1,0), 3*A1xB1(0,0) + A1xB1(0,2) + 2*A1xB1(1,2) + A1xB1(2,0), 2*A1xB1(0,1) + A1xB1(1,1)] EXAMPLES: (Automorphic type, including D4 triality) :: sage: [A3(chi).branch(A3,rule="automorphic") for chi in A3.fundamental_weights()] [A3(0,0,0,-1), A3(0,0,-1,-1), A3(0,-1,-1,-1)] sage: [D4(chi).branch(D4,rule="automorphic") for chi in D4.fundamental_weights()] [D4(1,0,0,0), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2)] sage: [D4(chi).branch(D4,rule="triality") for chi in D4.fundamental_weights()] [D4(1/2,1/2,1/2,-1/2), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1,0,0,0)] EXAMPLES: (Symmetric type) :: sage: [w.branch(B2,rule="symmetric") for w in [A4(1,0,0,0,0),A4(1,1,0,0,0),A4(1,1,1,0,0),A4(2,0,0,0,0)]] [B2(1,0), B2(1,1), B2(1,1), B2(0,0) + B2(2,0)] sage: [A5(w).branch(C3,rule="symmetric") for w in A5.fundamental_weights()] [C3(1,0,0), C3(0,0,0) + C3(1,1,0), C3(1,0,0) + C3(1,1,1), C3(0,0,0) + C3(1,1,0), C3(1,0,0)] sage: [A5(w).branch(D3,rule="symmetric") for w in A5.fundamental_weights()] [D3(1,0,0), D3(1,1,0), D3(1,1,-1) + D3(1,1,1), D3(1,1,0), D3(1,0,0)] sage: [D4(x).branch(B3,rule="symmetric") for x in D4.fundamental_weights()] [B3(0,0,0) + B3(1,0,0), B3(1,0,0) + B3(1,1,0), B3(1/2,1/2,1/2), B3(1/2,1/2,1/2)] sage: [D4(x).branch(G2,rule="symmetric") for x in D4.fundamental_weights()] [G2(0,0,0) + G2(1,0,-1), 2*G2(1,0,-1) + G2(2,-1,-1), G2(0,0,0) + G2(1,0,-1), G2(0,0,0) + G2(1,0,-1)] sage: [E6(fw).branch(F4,rule="symmetric") for fw in E6.fundamental_weights()] # long time (36s) [F4(0,0,0,0) + F4(0,0,0,1), F4(0,0,0,1) + F4(1,0,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(1,0,0,0) + 2*F4(0,0,1,0) + F4(1,0,0,1) + F4(0,1,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(0,0,0,0) + F4(0,0,0,1)] EXAMPLES: (Extended type) :: sage: [B3(x).branch(D3,rule="extended") for x in B3.fundamental_weights()] [D3(0,0,0) + D3(1,0,0), D3(1,0,0) + D3(1,1,0), D3(1/2,1/2,-1/2) + D3(1/2,1/2,1/2)] sage: [G2(w).branch(A2, rule="extended") for w in G2.fundamental_weights()] [A2(0,0,0) + A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3), A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3) + A2(1,0,-1)] sage: [F4(fw).branch(B4,rule="extended") for fw in F4.fundamental_weights()] # long time (9s) [B4(1/2,1/2,1/2,1/2) + B4(1,1,0,0), B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2) + B4(3/2,3/2,1/2,1/2) + B4(2,1,1,0), B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0) + B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2), B4(0,0,0,0) + B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0)] sage: E6 = WeylCharacterRing("E6", style="coroots") sage: A2xA2xA2=WeylCharacterRing("A2xA2xA2",style="coroots") sage: A5xA1=WeylCharacterRing("A5xA1",style="coroots") sage: G2 = WeylCharacterRing("G2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: A3xA1 = WeylCharacterRing("A3xA1", style="coroots") sage: A2xA2 = WeylCharacterRing("A2xA2", style="coroots") sage: A1xC3 = WeylCharacterRing("A1xC3",style="coroots") sage: E6(1,0,0,0,0,0).branch(A5xA1,rule="extended") # (0.7s) A5xA1(0,0,0,1,0,0) + A5xA1(1,0,0,0,0,1) sage: E6(1,0,0,0,0,0).branch(A2xA2xA2, rule="extended") # (0.7s) A2xA2xA2(0,0,0,1,1,0) + A2xA2xA2(0,1,1,0,0,0) + A2xA2xA2(1,0,0,0,0,1) sage: E7=WeylCharacterRing("E7",style="coroots") sage: A7=WeylCharacterRing("A7",style="coroots") sage: E7(1,0,0,0,0,0,0).branch(A7,rule="extended") # long time (5s) A7(0,0,0,1,0,0,0) + A7(1,0,0,0,0,0,1) sage: E8=WeylCharacterRing("E8",cache=true,style="coroots") sage: D8=WeylCharacterRing("D8",cache=true,style="coroots") sage: E8(0,0,0,0,0,0,0,1).branch(D8,rule="extended") # long time (19s) D8(0,0,0,0,0,0,1,0) + D8(0,1,0,0,0,0,0,0) sage: F4(1,0,0,0).branch(A1xC3,rule="extended") # (0.7s) A1xC3(0,2,0,0) + A1xC3(1,0,0,1) + A1xC3(2,0,0,0) sage: G2(0,1).branch(A1xA1, rule="extended") A1xA1(0,2) + A1xA1(2,0) + A1xA1(3,1) sage: F4(0,0,0,1).branch(A2xA2, rule="extended") # (0.4s) A2xA2(0,0,1,1) + A2xA2(0,1,0,1) + A2xA2(1,0,1,0) sage: F4(0,0,0,1).branch(A3xA1,rule="extended") # (0.34s) A3xA1(0,0,0,0) + A3xA1(0,0,0,2) + A3xA1(0,0,1,1) + A3xA1(0,1,0,0) + A3xA1(1,0,0,1) sage: D4=WeylCharacterRing("D4",style="coroots") sage: D2xD2=WeylCharacterRing("D2xD2",style="coroots") # We get D4 => A1xA1xA1xA1 by remembering that A1xA1 = D2. sage: [D4(fw).branch(D2xD2, rule="extended") for fw in D4.fundamental_weights()] [D2xD2(0,0,1,1) + D2xD2(1,1,0,0), D2xD2(0,0,2,0) + D2xD2(0,0,0,2) + D2xD2(2,0,0,0) + D2xD2(1,1,1,1) + D2xD2(0,2,0,0), D2xD2(1,0,0,1) + D2xD2(0,1,1,0), D2xD2(1,0,1,0) + D2xD2(0,1,0,1)] EXAMPLES: (Tensor type) :: sage: A5=WeylCharacterRing("A5", style="coroots") sage: A2xA1=WeylCharacterRing("A2xA1", style="coroots") sage: [A5(hwv).branch(A2xA1, rule="tensor") for hwv in A5.fundamental_weights()] [A2xA1(1,0,1), A2xA1(0,1,2) + A2xA1(2,0,0), A2xA1(0,0,3) + A2xA1(1,1,1), A2xA1(1,0,2) + A2xA1(0,2,0), A2xA1(0,1,1)] sage: B4=WeylCharacterRing("B4",style="coroots") sage: B1xB1=WeylCharacterRing("B1xB1",style="coroots") sage: [B4(f).branch(B1xB1,rule="tensor") for f in B4.fundamental_weights()] [B1xB1(2,2), B1xB1(0,2) + B1xB1(2,0) + B1xB1(2,4) + B1xB1(4,2), B1xB1(0,2) + B1xB1(0,6) + B1xB1(2,0) + B1xB1(2,2) + B1xB1(2,4) + B1xB1(4,2) + B1xB1(4,4) + B1xB1(6,0), B1xB1(1,3) + B1xB1(3,1)] sage: D4=WeylCharacterRing("D4",style="coroots") sage: C2xC1=WeylCharacterRing("C2xC1",style="coroots") sage: [D4(f).branch(C2xC1,rule="tensor") for f in D4.fundamental_weights()] [C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,2) + C2xC1(2,0,0), C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,0)] sage: C3=WeylCharacterRing("C3",style="coroots") sage: B1xC1=WeylCharacterRing("B1xC1",style="coroots") sage: [C3(f).branch(B1xC1,rule="tensor") for f in C3.fundamental_weights()] [B1xC1(2,1), B1xC1(2,2) + B1xC1(4,0), B1xC1(0,3) + B1xC1(4,1)] EXAMPLES: (Symmetric Power) :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: B3=WeylCharacterRing("B3",style="coroots") sage: C3=WeylCharacterRing("C3",style="coroots") sage: [B3(fw).branch(A1,rule="symmetric_power") for fw in B3.fundamental_weights()] [A1(6), A1(2) + A1(6) + A1(10), A1(0) + A1(6)] sage: [C3(fw).branch(A1,rule="symmetric_power") for fw in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] EXAMPLES: (Miscellaneous type) :: sage: G2 = WeylCharacterRing("G2") sage: [fw1, fw2, fw3] = B3.fundamental_weights() sage: B3(fw1+fw3).branch(G2, rule="miscellaneous") G2(1,0,-1) + G2(2,-1,-1) + G2(2,0,-2) EXAMPLES: (Isomorphic type) :: sage: [B2(x).branch(C2, rule="isomorphic") for x in B2.fundamental_weights()] [C2(1,1), C2(1,0)] sage: [C2(x).branch(B2, rule="isomorphic") for x in C2.fundamental_weights()] [B2(1/2,1/2), B2(1,0)] sage: [A3(x).branch(D3,rule="isomorphic") for x in A3.fundamental_weights()] [D3(1/2,1/2,1/2), D3(1,0,0), D3(1/2,1/2,-1/2)] sage: [D3(x).branch(A3,rule="isomorphic") for x in D3.fundamental_weights()] [A3(1/2,1/2,-1/2,-1/2), A3(1/4,1/4,1/4,-3/4), A3(3/4,-1/4,-1/4,-1/4)] Here A3(x,y,z,w) can be understood as a representation of SL(4). The weights x,y,z,w and x+t,y+t,z+t,w+t represent the same representation of SL(4) - though not of GL(4) - since A3(x+t,y+t,z+t,w+t) is the same as A3(x,y,z,w) tensored with `det^t`. So as a representation of SL(4), A3(1/4,1/4,1/4,-3/4) is the same as A3(1,1,1,0). The exterior square representation SL(4) -> GL(6) admits an invariant symmetric bilinear form, so is a representation SL(4) -> SO(6) that lifts to an isomorphism SL(4) -> Spin(6). Conversely, there are two isomorphisms SO(6) -> SL(4), of which we've selected one. In cases like this you might prefer style="coroots". :: sage: A3 = WeylCharacterRing("A3",style="coroots") sage: D3 = WeylCharacterRing("D3",style="coroots") sage: [D3(fw) for fw in D3.fundamental_weights()] [D3(1,0,0), D3(0,1,0), D3(0,0,1)] sage: [D3(fw).branch(A3,rule="isomorphic") for fw in D3.fundamental_weights()] [A3(0,1,0), A3(0,0,1), A3(1,0,0)] sage: D2 = WeylCharacterRing("D2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: [D2(fw).branch(A1xA1,rule="isomorphic") for fw in D2.fundamental_weights()] [A1xA1(1,0), A1xA1(0,1)] EXAMPLES: (Branching rules from plethysms) This is a general rule that includes any branching rule from types A,B,C or D as a special case. Thus it could be used in place of the above rules and would give the same results. However it is most useful when branching from G to a maximal subgroup H such that rank(H) < rank(G)-1. We consider a homomorphism H --> G where G is one of SL(r+1), SO(2r+1), Sp(2r) or SO(2r). The function branching_rule_from_plethysm produces the corresponding branching rule. The main ingredient is the character chi of the representation of H that is the homomorphism to GL(r+1), GL(2r+1) or GL(2r). This rule is so powerful that it contains the other rules implemented above as special cases. First let us consider the symmetric fifth power representation of SL(2). :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: chi=A1([5]) sage: chi.degree() 6 sage: chi.frobenius_schur_indicator() -1 This confirms that the character has degree 6 and is symplectic, so it corresponds to a homomorphism SL(2) --> Sp(6), and there is a corresponding branching rule C3 => A1. :: sage: C3 = WeylCharacterRing("C3",style="coroots") sage: sym5rule = branching_rule_from_plethysm(chi,"C3") sage: [C3(hwv).branch(A1,rule=sym5rule) for hwv in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] This is identical to the results we would obtain using rule="symmetric_power". The next example gives a branching not available by other standard rules. :: sage: G2 = WeylCharacterRing("G2",style="coroots") sage: D7 = WeylCharacterRing("D7",style="coroots") sage: ad=G2(0,1); ad.degree(); ad.frobenius_schur_indicator() 14 1 sage: spin = D7(0,0,0,0,0,1,0); spin.degree() 64 sage: spin.branch(G2, rule=branching_rule_from_plethysm(ad, "D7")) G2(1,1) We have confirmed that the adjoint representation of G2 gives a homomorphism into SO(14), and that the pullback of the one of the two 64 dimensional spin representations to SO(14) is an irreducible representation of G2. BRANCHING FROM A REDUCIBLE ROOT SYSTEM If you are branching from a reducible root system, the rule is a list of rules, one for each component type in the root system. The rules in the list are given in pairs [type, rule], where type is the root system to be branched to, and rule is the branching rule. :: sage: D4 = WeylCharacterRing("D4",style="coroots") sage: D2xD2 = WeylCharacterRing("D2xD2",style="coroots") sage: A1xA1xA1xA1 = WeylCharacterRing("A1xA1xA1xA1",style="coroots") sage: rr = [["A1xA1","isomorphic"],["A1xA1","isomorphic"]] sage: [D4(fw) for fw in D4.fundamental_weights()] [D4(1,0,0,0), D4(0,1,0,0), D4(0,0,1,0), D4(0,0,0,1)] sage: [D4(fw).branch(D2xD2,rule="extended").branch(A1xA1xA1xA1,rule=rr) for fw in D4.fundamental_weights()] [A1xA1xA1xA1(0,0,1,1) + A1xA1xA1xA1(1,1,0,0), A1xA1xA1xA1(0,0,0,2) + A1xA1xA1xA1(0,0,2,0) + A1xA1xA1xA1(0,2,0,0) + A1xA1xA1xA1(1,1,1,1) + A1xA1xA1xA1(2,0,0,0), A1xA1xA1xA1(0,1,1,0) + A1xA1xA1xA1(1,0,0,1), A1xA1xA1xA1(0,1,0,1) + A1xA1xA1xA1(1,0,1,0)] WRITING YOUR OWN RULES Suppose you want to branch from a group G to a subgroup H. Arrange the embedding so that a Cartan subalgebra U of H is contained in a Cartan subalgebra T of G. There is thus a mapping from the weight spaces Lie(T)* --> Lie(U)*. Two embeddings will produce identical branching rules if they differ by an element of the Weyl group of H. The RULE is this map Lie(T)* = G.space() to Lie(U)* = H.space(), which you may implement as a function. As an example, let us consider how to implement the branching rule A3 => C2. Here H = C2 = Sp(4) embedded as a subgroup in A3 = GL(4). The Cartan subalgebra U consists of diagonal matrices with eigenvalues u1, u2, -u2, -u1. The C2.space() is the two dimensional vector spaces consisting of the linear functionals u1 and u2 on U. On the other hand Lie(T) is RR^4. A convenient way to see the restriction is to think of it as the adjoint of the map [u1,u2] -> [u1,u2,-u2,-u1], that is, [x0,x1,x2,x3] -> [x0-x3,x1-x2]. Hence we may encode the rule: :: def rule(x): return [x[0]-x[3],x[1]-x[2]] or simply: :: rule = lambda x : [x[0]-x[3],x[1]-x[2]] EXAMPLES:: sage: A3 = WeylCharacterRing(['A',3]) sage: C2 = WeylCharacterRing(['C',2]) sage: rule = lambda x : [x[0]-x[3],x[1]-x[2]] sage: branch_weyl_character(A3([1,1,0,0]),A3,C2,rule) C2(0,0) + C2(1,1) sage: A3(1,1,0,0).branch(C2, rule) == C2(0,0) + C2(1,1) True """ if type(rule) == str: rule = get_branching_rule(R._cartan_type, S._cartan_type, rule) elif R._cartan_type.is_compound(): Rtypes = R._cartan_type.component_types() Stypes = [CartanType(l[0]) for l in rule] rules = [l[1] for l in rule] ntypes = len(Rtypes) rule_list = [get_branching_rule(Rtypes[i], Stypes[i], rules[i]) for i in range(ntypes)] shifts = R._cartan_type._shifts def rule(x): yl = [] for i in range(ntypes): yl.append(rule_list[i](x[shifts[i]:shifts[i+1]])) return flatten(yl) mdict = {} for k in chi._mdict: if S._style == "coroots": if S._cartan_type.is_atomic() and S._cartan_type[0] == 'E': if S._cartan_type[1] == 6: h = S._space(rule(list(k.to_vector()))) h = S.coerce_to_e6(h) elif S._cartan_type[1] == 7: h = S.coerce_to_e7(S._space(rule(list(k.to_vector())))) else: h = S.coerce_to_sl(S._space(rule(list(k.to_vector())))) else: h = S._space(rule(list(k.to_vector()))) if h in mdict: mdict[h] += chi._mdict[k] else: mdict[h] = chi._mdict[k] hdict = S.char_from_weights(mdict) return WeylCharacter(S, hdict, mdict)
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SYMMETRIC TYPE. Related to the automorphic type, when either the Dynkin diagram or the extended diagram has a symmetry there is a branching rule to the subalgebra (or subgroup) of invariants under the automorphism. Use rule="symmetric".
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SYMMETRIC TYPE. Related to the automorphic type, when G admits an outer automorphism (usually of degree 2) we may consider the branching rule to the isotropy subgroup H. In many cases the Dynkin diagram of H can be obtained by folding the Dynkin diagram of G. For such isotropy subgroups use rule="symmetric".
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def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Willenbring, Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601-1626, McKay and Patera, Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras (Marcel Dekker, 1981), and Fauser, Jarvis, King and Wybourne, New branching rules induced by plethysm. J. Phys. A 39 (2006), no. 11, 2611--2655. INPUT: - ``chi`` - a character of G - ``R`` - the Weyl Character Ring of G - ``S`` - the Weyl Character Ring of H - ``rule`` - a set of r dominant weights in H where r is the rank of G. You may use a predefined rule by specifying rule = one of"levi", "automorphic", "symmetric", "extended", "triality" or "miscellaneous". The use of these rules will be explained next. After the examples we will explain how to write your own branching rules for cases that we have omitted. To explain the predefined rules we survey the most important branching rules. These may be classified into several cases, and once this is understood, the detailed classification can be read off from the Dynkin diagrams. Dynkin classified the maximal subgroups of Lie groups in Mat. Sbornik N.S. 30(72):349-462 (1952). We will list give predefined rules that cover most cases where the branching rule is to a maximal subgroup. For convenience, we also give some branching rules to subgroups that are not maximal. For example, a Levi subgroup may or may not be maximal. LEVI TYPE. These can be read off from the Dynkin diagram. If removing a node from the Dynkin diagram produces another Dynkin diagram, there is a branching rule. Currently we require that the smaller diagram be connected. For these rules use the option rule="levi":: ['A',r] => ['A',r-1] ['B',r] => ['A',r-1] ['B',r] => ['B',r-1] ['C',r] => ['A',r-1] ['C',r] => ['C',r-1] ['D',r] => ['A',r-1] ['D',r] => ['D',r-1] ['E',r] => ['A',r-1] r = 7,8 ['E',r] => ['D',r-1] r = 6,7,8 ['E',r] => ['E',r-1] F4 => B3 F4 => C3 G2 => A1 (short root) Not all Levi subgroups are maximal subgroups. If the Levi is not maximal there may or may not be a preprogrammed rule="levi" for it. If there is not, the branching rule may still be obtained by going through an intermediate subgroup that is maximal using rule="extended". Thus the other Levi branching rule from G2 => A1 corresponding to the long root is available by first branching G2 => A_2 then A2 => A1. Similarly the branching rules to the Levi subgroup:: ['E',r] => ['A',r-1] r = 6,7,8 may be obtained by first branching E6=>A5xA1, E7=>A7 or E8=>A8. AUTOMORPHIC TYPE. If the Dynkin diagram has a symmetry, then there is an automorphism that is a special case of a branching rule. There is also an exotic "triality" automorphism of D4 having order 3. Use rule="automorphic" or (for D4) rule="triality":: ['A',r] => ['A',r] ['D',r] => ['D',r] E6 => E6 SYMMETRIC TYPE. Related to the automorphic type, when either the Dynkin diagram or the extended diagram has a symmetry there is a branching rule to the subalgebra (or subgroup) of invariants under the automorphism. Use rule="symmetric". The last branching rule, D4=>G2 is not to a maximal subgroup since D4=>B3=>G2, but it is included for convenience. :: ['A',2r+1] => ['B',r] ['A',2r] => ['C',r] ['A',2r] => ['D',r] ['D',r] => ['B',r-1] E6 => F4 D4 => G2 EXTENDED TYPE. If removing a node from the extended Dynkin diagram results in a Dynkin diagram, then there is a branching rule. Use rule="extended" for these. We will also use this classification for some rules that are not of this type, mainly involving type B, such as D6 => B3xB3. Here is the extended Dynkin diagram for D6:: 0 6 O O | | | | O---O---O---O---O 1 2 3 4 6 Removing the node 3 results in an embedding D3xD3 -> D6. This corresponds to the embedding SO(6)xSO(6) -> SO(12), and is of extended type. On the other hand the embedding SO(5)xSO(7)-->SO(12) (e.g. B2xB3 -> D6) cannot be explained this way but for uniformity is implemented under rule="extended". Using rule="extended" you can get any branching rule SO(n) => SO(a) x SO(b) x SO(c) x ... where n = a+b+c+ ... Sp(2n) => Sp(2a) x Sp(2b) x Sp(2c) x ... where n = a+b+c+ ... where O(a) = ['D',r] (a=2r) or ['B',r] (a=2r+1) and Sp(2r)=['C',r]. The following rules are implemented as special cases of rule="extended". :: E6 => A5xA1, A2xA2xA2 E7 => A7, D6xA1, A3xA3xA1 E8 => A8, D8, E7xA1, A4xA4, D5xA3, E6xA2 F4 => B4, C3xA1, A2xA2, A3xA1 G2 => A1xA1 Note that E8 has only a limited number of representations of reasonably low degree. TENSOR: There are branching rules: :: ['A', rs-1] => ['A',r-1] x ['A',s-1] ['B',2rs+r+s] => ['B',r] x ['B',s] ['D',2rs+s] => ['B',r] x ['D',s] ['D',2rs] => ['D',r] x ['D',s] ['D',2rs] => ['C',r] x ['C',s] ['C',2rs+s] => ['B',r] x ['C',s] ['C',2rs] => ['C',r] x ['D',s]. corresponding to the tensor product homomorphism. For type A, the homomorphism is GL(r) x GL(s) -> GL(rs). For the classical types, the relevant fact is that if V,W are orthogonal or symplectic spaces, that is, spaces endowed with symmetric or skew-symmetric bilinear forms, then V tensor W is also an orthogonal space (if V and W are both orthogonal or both symplectic) or symplectic (if one of V and W is orthogonal and the other symplectic). The corresponding branching rules are obtained using rule="tensor". SYMMETRIC POWER: The k-th symmetric and exterior power homomorphisms map GL(n) --> GL(binomial(n+k-1,k)) and GL(binomial(n,k)). The corresponding branching rules are not implemented but a special case is. The k-th symmetric power homomorphism SL(2) --> GL(k+1) has its image inside of SO(2r+1) if k=2r and inside of Sp(2r) if k=2r-1. Hence there are branching rules:: ['B',r] => A1 ['C',r] => A1 and these may be obtained using the rule "symmetric_power". MISCELLANEOUS: Use rule="miscellaneous" for the following rules:: B3 => G2 F4 => G2xA1 (not implemented yet) BRANCHING RULES FROM PLETHYSMS Nearly all branching rules G => H where G is of type A,B,C or D are covered by the preceding rules. The function branching_rules_from_plethysm covers the remaining cases. ISOMORPHIC TYPE: Although not usually referred to as a branching rule, the effects of the accidental isomorphisms may be handled using rule="isomorphic":: B2 => C2 C2 => B2 A3 => D3 D3 => A3 D2 => A1xA1 B1 => A1 C1 => A1 EXAMPLES: (Levi type) :: sage: A1 = WeylCharacterRing("A1") sage: A2 = WeylCharacterRing("A2") sage: A3 = WeylCharacterRing("A3") sage: A4 = WeylCharacterRing("A4") sage: A5 = WeylCharacterRing("A5") sage: B2 = WeylCharacterRing("B2") sage: B3 = WeylCharacterRing("B3") sage: B4 = WeylCharacterRing("B4") sage: C2 = WeylCharacterRing("C2") sage: C3 = WeylCharacterRing("C3") sage: D3 = WeylCharacterRing("D3") sage: D4 = WeylCharacterRing("D4") sage: D5 = WeylCharacterRing("D5") sage: G2 = WeylCharacterRing("G2") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: [B3(w).branch(A2,rule="levi") for w in B3.fundamental_weights()] [A2(0,0,-1) + A2(0,0,0) + A2(1,0,0), A2(0,-1,-1) + A2(0,0,-1) + A2(0,0,0) + A2(1,0,-1) + A2(1,0,0) + A2(1,1,0), A2(-1/2,-1/2,-1/2) + A2(1/2,-1/2,-1/2) + A2(1/2,1/2,-1/2) + A2(1/2,1/2,1/2)] The last example must be understood as follows. The representation of B3 being branched is spin, which is not a representation of SO(7) but of its double cover spin(7). The group A2 is really GL(3) and the double cover of SO(7) induces a cover of GL(3) that is trivial over SL(3) but not over the center of GL(3). The weight lattice for this GL(3) consists of triples (a,b,c) of half integers such that a-b and b-c are in `\ZZ`, and this is reflected in the last decomposition. :: sage: [C3(w).branch(A2,rule="levi") for w in C3.fundamental_weights()] [A2(0,0,-1) + A2(1,0,0), A2(0,-1,-1) + A2(1,0,-1) + A2(1,1,0), A2(-1,-1,-1) + A2(1,-1,-1) + A2(1,1,-1) + A2(1,1,1)] sage: [D4(w).branch(A3,rule="levi") for w in D4.fundamental_weights()] [A3(0,0,0,-1) + A3(1,0,0,0), A3(0,0,-1,-1) + A3(0,0,0,0) + A3(1,0,0,-1) + A3(1,1,0,0), A3(1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,-1/2), A3(-1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,1/2)] sage: [B3(w).branch(B2,rule="levi") for w in B3.fundamental_weights()] [2*B2(0,0) + B2(1,0), B2(0,0) + 2*B2(1,0) + B2(1,1), 2*B2(1/2,1/2)] sage: C3 = WeylCharacterRing(['C',3]) sage: [C3(w).branch(C2,rule="levi") for w in C3.fundamental_weights()] [2*C2(0,0) + C2(1,0), C2(0,0) + 2*C2(1,0) + C2(1,1), C2(1,0) + 2*C2(1,1)] sage: [D5(w).branch(D4,rule="levi") for w in D5.fundamental_weights()] [2*D4(0,0,0,0) + D4(1,0,0,0), D4(0,0,0,0) + 2*D4(1,0,0,0) + D4(1,1,0,0), D4(1,0,0,0) + 2*D4(1,1,0,0) + D4(1,1,1,0), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2)] sage: G2(1,0,-1).branch(A1,rule="levi") A1(0,-1) + A1(1,-1) + A1(1,0) sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: fw = E6.fundamental_weights() sage: [E6(fw[i]).branch(D5,rule="levi") for i in [1,2,6]] # long time (3s) [D5(0,0,0,0,0) + D5(0,0,0,0,1) + D5(1,0,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(0,0,0,0,1) + D5(0,1,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(1,0,0,0,0)] sage: E7=WeylCharacterRing("E7",style="coroots") sage: D6=WeylCharacterRing("D6",style="coroots") sage: fw = E7.fundamental_weights() sage: [E7(fw[i]).branch(D6,rule="levi") for i in [1,2,7]] # long time (26s) [3*D6(0,0,0,0,0,0) + 2*D6(0,0,0,0,1,0) + D6(0,1,0,0,0,0), 3*D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0) + 2*D6(0,0,1,0,0,0) + D6(1,0,0,0,1,0), D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0)] sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8=WeylCharacterRing("E8",style="coroots",cache=True) sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8(1,0,0,0,0,0,0,0).branch(D7,rule="levi") # not tested (very long time) (160s) 3*D7(0,0,0,0,0,0,0) + 2*D7(0,0,0,0,0,1,0) + 2*D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) + 2*D7(0,0,1,0,0,0,0) + D7(0,0,0,1,0,0,0) + D7(1,0,0,0,0,1,0) + D7(1,0,0,0,0,0,1) + D7(2,0,0,0,0,0,0) sage: E8(0,0,0,0,0,0,0,1).branch(D7,rule="levi") # long time (3s) D7(0,0,0,0,0,0,0) + D7(0,0,0,0,0,1,0) + D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) sage: [F4(fw).branch(B3,rule="levi") for fw in F4.fundamental_weights()] # long time (36s) [B3(0,0,0) + 2*B3(1/2,1/2,1/2) + 2*B3(1,0,0) + B3(1,1,0), B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 5*B3(1,0,0) + 7*B3(1,1,0) + 3*B3(1,1,1) + 6*B3(3/2,1/2,1/2) + 2*B3(3/2,3/2,1/2) + B3(2,0,0) + 2*B3(2,1,0) + B3(2,1,1), 3*B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 4*B3(1,0,0) + 3*B3(1,1,0) + B3(1,1,1) + 2*B3(3/2,1/2,1/2), 3*B3(0,0,0) + 2*B3(1/2,1/2,1/2) + B3(1,0,0)] sage: [F4(fw).branch(C3,rule="levi") for fw in F4.fundamental_weights()] # long time (6s) [3*C3(0,0,0) + 2*C3(1,1,1) + C3(2,0,0), 3*C3(0,0,0) + 6*C3(1,1,1) + 4*C3(2,0,0) + 2*C3(2,1,0) + 3*C3(2,2,0) + C3(2,2,2) + C3(3,1,0) + 2*C3(3,1,1), 2*C3(1,0,0) + 3*C3(1,1,0) + C3(2,0,0) + 2*C3(2,1,0) + C3(2,1,1), 2*C3(1,0,0) + C3(1,1,0)] sage: A1xA1 = WeylCharacterRing("A1xA1") sage: [A3(hwv).branch(A1xA1,rule="levi") for hwv in A3.fundamental_weights()] [A1xA1(0,0,1,0) + A1xA1(1,0,0,0), A1xA1(0,0,1,1) + A1xA1(1,0,1,0) + A1xA1(1,1,0,0), A1xA1(1,0,1,1) + A1xA1(1,1,1,0)] sage: A1xB1=WeylCharacterRing("A1xB1",style="coroots") sage: [B3(x).branch(A1xB1,rule="levi") for x in B3.fundamental_weights()] [A1xB1(0,2) + 2*A1xB1(1,0), 3*A1xB1(0,0) + A1xB1(0,2) + 2*A1xB1(1,2) + A1xB1(2,0), 2*A1xB1(0,1) + A1xB1(1,1)] EXAMPLES: (Automorphic type, including D4 triality) :: sage: [A3(chi).branch(A3,rule="automorphic") for chi in A3.fundamental_weights()] [A3(0,0,0,-1), A3(0,0,-1,-1), A3(0,-1,-1,-1)] sage: [D4(chi).branch(D4,rule="automorphic") for chi in D4.fundamental_weights()] [D4(1,0,0,0), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2)] sage: [D4(chi).branch(D4,rule="triality") for chi in D4.fundamental_weights()] [D4(1/2,1/2,1/2,-1/2), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1,0,0,0)] EXAMPLES: (Symmetric type) :: sage: [w.branch(B2,rule="symmetric") for w in [A4(1,0,0,0,0),A4(1,1,0,0,0),A4(1,1,1,0,0),A4(2,0,0,0,0)]] [B2(1,0), B2(1,1), B2(1,1), B2(0,0) + B2(2,0)] sage: [A5(w).branch(C3,rule="symmetric") for w in A5.fundamental_weights()] [C3(1,0,0), C3(0,0,0) + C3(1,1,0), C3(1,0,0) + C3(1,1,1), C3(0,0,0) + C3(1,1,0), C3(1,0,0)] sage: [A5(w).branch(D3,rule="symmetric") for w in A5.fundamental_weights()] [D3(1,0,0), D3(1,1,0), D3(1,1,-1) + D3(1,1,1), D3(1,1,0), D3(1,0,0)] sage: [D4(x).branch(B3,rule="symmetric") for x in D4.fundamental_weights()] [B3(0,0,0) + B3(1,0,0), B3(1,0,0) + B3(1,1,0), B3(1/2,1/2,1/2), B3(1/2,1/2,1/2)] sage: [D4(x).branch(G2,rule="symmetric") for x in D4.fundamental_weights()] [G2(0,0,0) + G2(1,0,-1), 2*G2(1,0,-1) + G2(2,-1,-1), G2(0,0,0) + G2(1,0,-1), G2(0,0,0) + G2(1,0,-1)] sage: [E6(fw).branch(F4,rule="symmetric") for fw in E6.fundamental_weights()] # long time (36s) [F4(0,0,0,0) + F4(0,0,0,1), F4(0,0,0,1) + F4(1,0,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(1,0,0,0) + 2*F4(0,0,1,0) + F4(1,0,0,1) + F4(0,1,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(0,0,0,0) + F4(0,0,0,1)] EXAMPLES: (Extended type) :: sage: [B3(x).branch(D3,rule="extended") for x in B3.fundamental_weights()] [D3(0,0,0) + D3(1,0,0), D3(1,0,0) + D3(1,1,0), D3(1/2,1/2,-1/2) + D3(1/2,1/2,1/2)] sage: [G2(w).branch(A2, rule="extended") for w in G2.fundamental_weights()] [A2(0,0,0) + A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3), A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3) + A2(1,0,-1)] sage: [F4(fw).branch(B4,rule="extended") for fw in F4.fundamental_weights()] # long time (9s) [B4(1/2,1/2,1/2,1/2) + B4(1,1,0,0), B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2) + B4(3/2,3/2,1/2,1/2) + B4(2,1,1,0), B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0) + B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2), B4(0,0,0,0) + B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0)] sage: E6 = WeylCharacterRing("E6", style="coroots") sage: A2xA2xA2=WeylCharacterRing("A2xA2xA2",style="coroots") sage: A5xA1=WeylCharacterRing("A5xA1",style="coroots") sage: G2 = WeylCharacterRing("G2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: A3xA1 = WeylCharacterRing("A3xA1", style="coroots") sage: A2xA2 = WeylCharacterRing("A2xA2", style="coroots") sage: A1xC3 = WeylCharacterRing("A1xC3",style="coroots") sage: E6(1,0,0,0,0,0).branch(A5xA1,rule="extended") # (0.7s) A5xA1(0,0,0,1,0,0) + A5xA1(1,0,0,0,0,1) sage: E6(1,0,0,0,0,0).branch(A2xA2xA2, rule="extended") # (0.7s) A2xA2xA2(0,0,0,1,1,0) + A2xA2xA2(0,1,1,0,0,0) + A2xA2xA2(1,0,0,0,0,1) sage: E7=WeylCharacterRing("E7",style="coroots") sage: A7=WeylCharacterRing("A7",style="coroots") sage: E7(1,0,0,0,0,0,0).branch(A7,rule="extended") # long time (5s) A7(0,0,0,1,0,0,0) + A7(1,0,0,0,0,0,1) sage: E8=WeylCharacterRing("E8",cache=true,style="coroots") sage: D8=WeylCharacterRing("D8",cache=true,style="coroots") sage: E8(0,0,0,0,0,0,0,1).branch(D8,rule="extended") # long time (19s) D8(0,0,0,0,0,0,1,0) + D8(0,1,0,0,0,0,0,0) sage: F4(1,0,0,0).branch(A1xC3,rule="extended") # (0.7s) A1xC3(0,2,0,0) + A1xC3(1,0,0,1) + A1xC3(2,0,0,0) sage: G2(0,1).branch(A1xA1, rule="extended") A1xA1(0,2) + A1xA1(2,0) + A1xA1(3,1) sage: F4(0,0,0,1).branch(A2xA2, rule="extended") # (0.4s) A2xA2(0,0,1,1) + A2xA2(0,1,0,1) + A2xA2(1,0,1,0) sage: F4(0,0,0,1).branch(A3xA1,rule="extended") # (0.34s) A3xA1(0,0,0,0) + A3xA1(0,0,0,2) + A3xA1(0,0,1,1) + A3xA1(0,1,0,0) + A3xA1(1,0,0,1) sage: D4=WeylCharacterRing("D4",style="coroots") sage: D2xD2=WeylCharacterRing("D2xD2",style="coroots") # We get D4 => A1xA1xA1xA1 by remembering that A1xA1 = D2. sage: [D4(fw).branch(D2xD2, rule="extended") for fw in D4.fundamental_weights()] [D2xD2(0,0,1,1) + D2xD2(1,1,0,0), D2xD2(0,0,2,0) + D2xD2(0,0,0,2) + D2xD2(2,0,0,0) + D2xD2(1,1,1,1) + D2xD2(0,2,0,0), D2xD2(1,0,0,1) + D2xD2(0,1,1,0), D2xD2(1,0,1,0) + D2xD2(0,1,0,1)] EXAMPLES: (Tensor type) :: sage: A5=WeylCharacterRing("A5", style="coroots") sage: A2xA1=WeylCharacterRing("A2xA1", style="coroots") sage: [A5(hwv).branch(A2xA1, rule="tensor") for hwv in A5.fundamental_weights()] [A2xA1(1,0,1), A2xA1(0,1,2) + A2xA1(2,0,0), A2xA1(0,0,3) + A2xA1(1,1,1), A2xA1(1,0,2) + A2xA1(0,2,0), A2xA1(0,1,1)] sage: B4=WeylCharacterRing("B4",style="coroots") sage: B1xB1=WeylCharacterRing("B1xB1",style="coroots") sage: [B4(f).branch(B1xB1,rule="tensor") for f in B4.fundamental_weights()] [B1xB1(2,2), B1xB1(0,2) + B1xB1(2,0) + B1xB1(2,4) + B1xB1(4,2), B1xB1(0,2) + B1xB1(0,6) + B1xB1(2,0) + B1xB1(2,2) + B1xB1(2,4) + B1xB1(4,2) + B1xB1(4,4) + B1xB1(6,0), B1xB1(1,3) + B1xB1(3,1)] sage: D4=WeylCharacterRing("D4",style="coroots") sage: C2xC1=WeylCharacterRing("C2xC1",style="coroots") sage: [D4(f).branch(C2xC1,rule="tensor") for f in D4.fundamental_weights()] [C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,2) + C2xC1(2,0,0), C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,0)] sage: C3=WeylCharacterRing("C3",style="coroots") sage: B1xC1=WeylCharacterRing("B1xC1",style="coroots") sage: [C3(f).branch(B1xC1,rule="tensor") for f in C3.fundamental_weights()] [B1xC1(2,1), B1xC1(2,2) + B1xC1(4,0), B1xC1(0,3) + B1xC1(4,1)] EXAMPLES: (Symmetric Power) :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: B3=WeylCharacterRing("B3",style="coroots") sage: C3=WeylCharacterRing("C3",style="coroots") sage: [B3(fw).branch(A1,rule="symmetric_power") for fw in B3.fundamental_weights()] [A1(6), A1(2) + A1(6) + A1(10), A1(0) + A1(6)] sage: [C3(fw).branch(A1,rule="symmetric_power") for fw in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] EXAMPLES: (Miscellaneous type) :: sage: G2 = WeylCharacterRing("G2") sage: [fw1, fw2, fw3] = B3.fundamental_weights() sage: B3(fw1+fw3).branch(G2, rule="miscellaneous") G2(1,0,-1) + G2(2,-1,-1) + G2(2,0,-2) EXAMPLES: (Isomorphic type) :: sage: [B2(x).branch(C2, rule="isomorphic") for x in B2.fundamental_weights()] [C2(1,1), C2(1,0)] sage: [C2(x).branch(B2, rule="isomorphic") for x in C2.fundamental_weights()] [B2(1/2,1/2), B2(1,0)] sage: [A3(x).branch(D3,rule="isomorphic") for x in A3.fundamental_weights()] [D3(1/2,1/2,1/2), D3(1,0,0), D3(1/2,1/2,-1/2)] sage: [D3(x).branch(A3,rule="isomorphic") for x in D3.fundamental_weights()] [A3(1/2,1/2,-1/2,-1/2), A3(1/4,1/4,1/4,-3/4), A3(3/4,-1/4,-1/4,-1/4)] Here A3(x,y,z,w) can be understood as a representation of SL(4). The weights x,y,z,w and x+t,y+t,z+t,w+t represent the same representation of SL(4) - though not of GL(4) - since A3(x+t,y+t,z+t,w+t) is the same as A3(x,y,z,w) tensored with `det^t`. So as a representation of SL(4), A3(1/4,1/4,1/4,-3/4) is the same as A3(1,1,1,0). The exterior square representation SL(4) -> GL(6) admits an invariant symmetric bilinear form, so is a representation SL(4) -> SO(6) that lifts to an isomorphism SL(4) -> Spin(6). Conversely, there are two isomorphisms SO(6) -> SL(4), of which we've selected one. In cases like this you might prefer style="coroots". :: sage: A3 = WeylCharacterRing("A3",style="coroots") sage: D3 = WeylCharacterRing("D3",style="coroots") sage: [D3(fw) for fw in D3.fundamental_weights()] [D3(1,0,0), D3(0,1,0), D3(0,0,1)] sage: [D3(fw).branch(A3,rule="isomorphic") for fw in D3.fundamental_weights()] [A3(0,1,0), A3(0,0,1), A3(1,0,0)] sage: D2 = WeylCharacterRing("D2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: [D2(fw).branch(A1xA1,rule="isomorphic") for fw in D2.fundamental_weights()] [A1xA1(1,0), A1xA1(0,1)] EXAMPLES: (Branching rules from plethysms) This is a general rule that includes any branching rule from types A,B,C or D as a special case. Thus it could be used in place of the above rules and would give the same results. However it is most useful when branching from G to a maximal subgroup H such that rank(H) < rank(G)-1. We consider a homomorphism H --> G where G is one of SL(r+1), SO(2r+1), Sp(2r) or SO(2r). The function branching_rule_from_plethysm produces the corresponding branching rule. The main ingredient is the character chi of the representation of H that is the homomorphism to GL(r+1), GL(2r+1) or GL(2r). This rule is so powerful that it contains the other rules implemented above as special cases. First let us consider the symmetric fifth power representation of SL(2). :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: chi=A1([5]) sage: chi.degree() 6 sage: chi.frobenius_schur_indicator() -1 This confirms that the character has degree 6 and is symplectic, so it corresponds to a homomorphism SL(2) --> Sp(6), and there is a corresponding branching rule C3 => A1. :: sage: C3 = WeylCharacterRing("C3",style="coroots") sage: sym5rule = branching_rule_from_plethysm(chi,"C3") sage: [C3(hwv).branch(A1,rule=sym5rule) for hwv in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] This is identical to the results we would obtain using rule="symmetric_power". The next example gives a branching not available by other standard rules. :: sage: G2 = WeylCharacterRing("G2",style="coroots") sage: D7 = WeylCharacterRing("D7",style="coroots") sage: ad=G2(0,1); ad.degree(); ad.frobenius_schur_indicator() 14 1 sage: spin = D7(0,0,0,0,0,1,0); spin.degree() 64 sage: spin.branch(G2, rule=branching_rule_from_plethysm(ad, "D7")) G2(1,1) We have confirmed that the adjoint representation of G2 gives a homomorphism into SO(14), and that the pullback of the one of the two 64 dimensional spin representations to SO(14) is an irreducible representation of G2. BRANCHING FROM A REDUCIBLE ROOT SYSTEM If you are branching from a reducible root system, the rule is a list of rules, one for each component type in the root system. The rules in the list are given in pairs [type, rule], where type is the root system to be branched to, and rule is the branching rule. :: sage: D4 = WeylCharacterRing("D4",style="coroots") sage: D2xD2 = WeylCharacterRing("D2xD2",style="coroots") sage: A1xA1xA1xA1 = WeylCharacterRing("A1xA1xA1xA1",style="coroots") sage: rr = [["A1xA1","isomorphic"],["A1xA1","isomorphic"]] sage: [D4(fw) for fw in D4.fundamental_weights()] [D4(1,0,0,0), D4(0,1,0,0), D4(0,0,1,0), D4(0,0,0,1)] sage: [D4(fw).branch(D2xD2,rule="extended").branch(A1xA1xA1xA1,rule=rr) for fw in D4.fundamental_weights()] [A1xA1xA1xA1(0,0,1,1) + A1xA1xA1xA1(1,1,0,0), A1xA1xA1xA1(0,0,0,2) + A1xA1xA1xA1(0,0,2,0) + A1xA1xA1xA1(0,2,0,0) + A1xA1xA1xA1(1,1,1,1) + A1xA1xA1xA1(2,0,0,0), A1xA1xA1xA1(0,1,1,0) + A1xA1xA1xA1(1,0,0,1), A1xA1xA1xA1(0,1,0,1) + A1xA1xA1xA1(1,0,1,0)] WRITING YOUR OWN RULES Suppose you want to branch from a group G to a subgroup H. Arrange the embedding so that a Cartan subalgebra U of H is contained in a Cartan subalgebra T of G. There is thus a mapping from the weight spaces Lie(T)* --> Lie(U)*. Two embeddings will produce identical branching rules if they differ by an element of the Weyl group of H. The RULE is this map Lie(T)* = G.space() to Lie(U)* = H.space(), which you may implement as a function. As an example, let us consider how to implement the branching rule A3 => C2. Here H = C2 = Sp(4) embedded as a subgroup in A3 = GL(4). The Cartan subalgebra U consists of diagonal matrices with eigenvalues u1, u2, -u2, -u1. The C2.space() is the two dimensional vector spaces consisting of the linear functionals u1 and u2 on U. On the other hand Lie(T) is RR^4. A convenient way to see the restriction is to think of it as the adjoint of the map [u1,u2] -> [u1,u2,-u2,-u1], that is, [x0,x1,x2,x3] -> [x0-x3,x1-x2]. Hence we may encode the rule: :: def rule(x): return [x[0]-x[3],x[1]-x[2]] or simply: :: rule = lambda x : [x[0]-x[3],x[1]-x[2]] EXAMPLES:: sage: A3 = WeylCharacterRing(['A',3]) sage: C2 = WeylCharacterRing(['C',2]) sage: rule = lambda x : [x[0]-x[3],x[1]-x[2]] sage: branch_weyl_character(A3([1,1,0,0]),A3,C2,rule) C2(0,0) + C2(1,1) sage: A3(1,1,0,0).branch(C2, rule) == C2(0,0) + C2(1,1) True """ if type(rule) == str: rule = get_branching_rule(R._cartan_type, S._cartan_type, rule) elif R._cartan_type.is_compound(): Rtypes = R._cartan_type.component_types() Stypes = [CartanType(l[0]) for l in rule] rules = [l[1] for l in rule] ntypes = len(Rtypes) rule_list = [get_branching_rule(Rtypes[i], Stypes[i], rules[i]) for i in range(ntypes)] shifts = R._cartan_type._shifts def rule(x): yl = [] for i in range(ntypes): yl.append(rule_list[i](x[shifts[i]:shifts[i+1]])) return flatten(yl) mdict = {} for k in chi._mdict: if S._style == "coroots": if S._cartan_type.is_atomic() and S._cartan_type[0] == 'E': if S._cartan_type[1] == 6: h = S._space(rule(list(k.to_vector()))) h = S.coerce_to_e6(h) elif S._cartan_type[1] == 7: h = S.coerce_to_e7(S._space(rule(list(k.to_vector())))) else: h = S.coerce_to_sl(S._space(rule(list(k.to_vector())))) else: h = S._space(rule(list(k.to_vector()))) if h in mdict: mdict[h] += chi._mdict[k] else: mdict[h] = chi._mdict[k] hdict = S.char_from_weights(mdict) return WeylCharacter(S, hdict, mdict)
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Using rule="extended" you can get any branching rule SO(n) => SO(a) x SO(b) x SO(c) x ... where n = a+b+c+ ... Sp(2n) => Sp(2a) x Sp(2b) x Sp(2c) x ... where n = a+b+c+ ... where O(a) = ['D',r] (a=2r) or ['B',r] (a=2r+1) and Sp(2r)=['C',r].
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def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Willenbring, Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601-1626, McKay and Patera, Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras (Marcel Dekker, 1981), and Fauser, Jarvis, King and Wybourne, New branching rules induced by plethysm. J. Phys. A 39 (2006), no. 11, 2611--2655. INPUT: - ``chi`` - a character of G - ``R`` - the Weyl Character Ring of G - ``S`` - the Weyl Character Ring of H - ``rule`` - a set of r dominant weights in H where r is the rank of G. You may use a predefined rule by specifying rule = one of"levi", "automorphic", "symmetric", "extended", "triality" or "miscellaneous". The use of these rules will be explained next. After the examples we will explain how to write your own branching rules for cases that we have omitted. To explain the predefined rules we survey the most important branching rules. These may be classified into several cases, and once this is understood, the detailed classification can be read off from the Dynkin diagrams. Dynkin classified the maximal subgroups of Lie groups in Mat. Sbornik N.S. 30(72):349-462 (1952). We will list give predefined rules that cover most cases where the branching rule is to a maximal subgroup. For convenience, we also give some branching rules to subgroups that are not maximal. For example, a Levi subgroup may or may not be maximal. LEVI TYPE. These can be read off from the Dynkin diagram. If removing a node from the Dynkin diagram produces another Dynkin diagram, there is a branching rule. Currently we require that the smaller diagram be connected. For these rules use the option rule="levi":: ['A',r] => ['A',r-1] ['B',r] => ['A',r-1] ['B',r] => ['B',r-1] ['C',r] => ['A',r-1] ['C',r] => ['C',r-1] ['D',r] => ['A',r-1] ['D',r] => ['D',r-1] ['E',r] => ['A',r-1] r = 7,8 ['E',r] => ['D',r-1] r = 6,7,8 ['E',r] => ['E',r-1] F4 => B3 F4 => C3 G2 => A1 (short root) Not all Levi subgroups are maximal subgroups. If the Levi is not maximal there may or may not be a preprogrammed rule="levi" for it. If there is not, the branching rule may still be obtained by going through an intermediate subgroup that is maximal using rule="extended". Thus the other Levi branching rule from G2 => A1 corresponding to the long root is available by first branching G2 => A_2 then A2 => A1. Similarly the branching rules to the Levi subgroup:: ['E',r] => ['A',r-1] r = 6,7,8 may be obtained by first branching E6=>A5xA1, E7=>A7 or E8=>A8. AUTOMORPHIC TYPE. If the Dynkin diagram has a symmetry, then there is an automorphism that is a special case of a branching rule. There is also an exotic "triality" automorphism of D4 having order 3. Use rule="automorphic" or (for D4) rule="triality":: ['A',r] => ['A',r] ['D',r] => ['D',r] E6 => E6 SYMMETRIC TYPE. Related to the automorphic type, when either the Dynkin diagram or the extended diagram has a symmetry there is a branching rule to the subalgebra (or subgroup) of invariants under the automorphism. Use rule="symmetric". The last branching rule, D4=>G2 is not to a maximal subgroup since D4=>B3=>G2, but it is included for convenience. :: ['A',2r+1] => ['B',r] ['A',2r] => ['C',r] ['A',2r] => ['D',r] ['D',r] => ['B',r-1] E6 => F4 D4 => G2 EXTENDED TYPE. If removing a node from the extended Dynkin diagram results in a Dynkin diagram, then there is a branching rule. Use rule="extended" for these. We will also use this classification for some rules that are not of this type, mainly involving type B, such as D6 => B3xB3. Here is the extended Dynkin diagram for D6:: 0 6 O O | | | | O---O---O---O---O 1 2 3 4 6 Removing the node 3 results in an embedding D3xD3 -> D6. This corresponds to the embedding SO(6)xSO(6) -> SO(12), and is of extended type. On the other hand the embedding SO(5)xSO(7)-->SO(12) (e.g. B2xB3 -> D6) cannot be explained this way but for uniformity is implemented under rule="extended". Using rule="extended" you can get any branching rule SO(n) => SO(a) x SO(b) x SO(c) x ... where n = a+b+c+ ... Sp(2n) => Sp(2a) x Sp(2b) x Sp(2c) x ... where n = a+b+c+ ... where O(a) = ['D',r] (a=2r) or ['B',r] (a=2r+1) and Sp(2r)=['C',r]. The following rules are implemented as special cases of rule="extended". :: E6 => A5xA1, A2xA2xA2 E7 => A7, D6xA1, A3xA3xA1 E8 => A8, D8, E7xA1, A4xA4, D5xA3, E6xA2 F4 => B4, C3xA1, A2xA2, A3xA1 G2 => A1xA1 Note that E8 has only a limited number of representations of reasonably low degree. TENSOR: There are branching rules: :: ['A', rs-1] => ['A',r-1] x ['A',s-1] ['B',2rs+r+s] => ['B',r] x ['B',s] ['D',2rs+s] => ['B',r] x ['D',s] ['D',2rs] => ['D',r] x ['D',s] ['D',2rs] => ['C',r] x ['C',s] ['C',2rs+s] => ['B',r] x ['C',s] ['C',2rs] => ['C',r] x ['D',s]. corresponding to the tensor product homomorphism. For type A, the homomorphism is GL(r) x GL(s) -> GL(rs). For the classical types, the relevant fact is that if V,W are orthogonal or symplectic spaces, that is, spaces endowed with symmetric or skew-symmetric bilinear forms, then V tensor W is also an orthogonal space (if V and W are both orthogonal or both symplectic) or symplectic (if one of V and W is orthogonal and the other symplectic). The corresponding branching rules are obtained using rule="tensor". SYMMETRIC POWER: The k-th symmetric and exterior power homomorphisms map GL(n) --> GL(binomial(n+k-1,k)) and GL(binomial(n,k)). The corresponding branching rules are not implemented but a special case is. The k-th symmetric power homomorphism SL(2) --> GL(k+1) has its image inside of SO(2r+1) if k=2r and inside of Sp(2r) if k=2r-1. Hence there are branching rules:: ['B',r] => A1 ['C',r] => A1 and these may be obtained using the rule "symmetric_power". MISCELLANEOUS: Use rule="miscellaneous" for the following rules:: B3 => G2 F4 => G2xA1 (not implemented yet) BRANCHING RULES FROM PLETHYSMS Nearly all branching rules G => H where G is of type A,B,C or D are covered by the preceding rules. The function branching_rules_from_plethysm covers the remaining cases. ISOMORPHIC TYPE: Although not usually referred to as a branching rule, the effects of the accidental isomorphisms may be handled using rule="isomorphic":: B2 => C2 C2 => B2 A3 => D3 D3 => A3 D2 => A1xA1 B1 => A1 C1 => A1 EXAMPLES: (Levi type) :: sage: A1 = WeylCharacterRing("A1") sage: A2 = WeylCharacterRing("A2") sage: A3 = WeylCharacterRing("A3") sage: A4 = WeylCharacterRing("A4") sage: A5 = WeylCharacterRing("A5") sage: B2 = WeylCharacterRing("B2") sage: B3 = WeylCharacterRing("B3") sage: B4 = WeylCharacterRing("B4") sage: C2 = WeylCharacterRing("C2") sage: C3 = WeylCharacterRing("C3") sage: D3 = WeylCharacterRing("D3") sage: D4 = WeylCharacterRing("D4") sage: D5 = WeylCharacterRing("D5") sage: G2 = WeylCharacterRing("G2") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: [B3(w).branch(A2,rule="levi") for w in B3.fundamental_weights()] [A2(0,0,-1) + A2(0,0,0) + A2(1,0,0), A2(0,-1,-1) + A2(0,0,-1) + A2(0,0,0) + A2(1,0,-1) + A2(1,0,0) + A2(1,1,0), A2(-1/2,-1/2,-1/2) + A2(1/2,-1/2,-1/2) + A2(1/2,1/2,-1/2) + A2(1/2,1/2,1/2)] The last example must be understood as follows. The representation of B3 being branched is spin, which is not a representation of SO(7) but of its double cover spin(7). The group A2 is really GL(3) and the double cover of SO(7) induces a cover of GL(3) that is trivial over SL(3) but not over the center of GL(3). The weight lattice for this GL(3) consists of triples (a,b,c) of half integers such that a-b and b-c are in `\ZZ`, and this is reflected in the last decomposition. :: sage: [C3(w).branch(A2,rule="levi") for w in C3.fundamental_weights()] [A2(0,0,-1) + A2(1,0,0), A2(0,-1,-1) + A2(1,0,-1) + A2(1,1,0), A2(-1,-1,-1) + A2(1,-1,-1) + A2(1,1,-1) + A2(1,1,1)] sage: [D4(w).branch(A3,rule="levi") for w in D4.fundamental_weights()] [A3(0,0,0,-1) + A3(1,0,0,0), A3(0,0,-1,-1) + A3(0,0,0,0) + A3(1,0,0,-1) + A3(1,1,0,0), A3(1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,-1/2), A3(-1/2,-1/2,-1/2,-1/2) + A3(1/2,1/2,-1/2,-1/2) + A3(1/2,1/2,1/2,1/2)] sage: [B3(w).branch(B2,rule="levi") for w in B3.fundamental_weights()] [2*B2(0,0) + B2(1,0), B2(0,0) + 2*B2(1,0) + B2(1,1), 2*B2(1/2,1/2)] sage: C3 = WeylCharacterRing(['C',3]) sage: [C3(w).branch(C2,rule="levi") for w in C3.fundamental_weights()] [2*C2(0,0) + C2(1,0), C2(0,0) + 2*C2(1,0) + C2(1,1), C2(1,0) + 2*C2(1,1)] sage: [D5(w).branch(D4,rule="levi") for w in D5.fundamental_weights()] [2*D4(0,0,0,0) + D4(1,0,0,0), D4(0,0,0,0) + 2*D4(1,0,0,0) + D4(1,1,0,0), D4(1,0,0,0) + 2*D4(1,1,0,0) + D4(1,1,1,0), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2) + D4(1/2,1/2,1/2,1/2)] sage: G2(1,0,-1).branch(A1,rule="levi") A1(0,-1) + A1(1,-1) + A1(1,0) sage: E6=WeylCharacterRing("E6",style="coroots") sage: D5=WeylCharacterRing("D5",style="coroots") sage: fw = E6.fundamental_weights() sage: [E6(fw[i]).branch(D5,rule="levi") for i in [1,2,6]] # long time (3s) [D5(0,0,0,0,0) + D5(0,0,0,0,1) + D5(1,0,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(0,0,0,0,1) + D5(0,1,0,0,0), D5(0,0,0,0,0) + D5(0,0,0,1,0) + D5(1,0,0,0,0)] sage: E7=WeylCharacterRing("E7",style="coroots") sage: D6=WeylCharacterRing("D6",style="coroots") sage: fw = E7.fundamental_weights() sage: [E7(fw[i]).branch(D6,rule="levi") for i in [1,2,7]] # long time (26s) [3*D6(0,0,0,0,0,0) + 2*D6(0,0,0,0,1,0) + D6(0,1,0,0,0,0), 3*D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0) + 2*D6(0,0,1,0,0,0) + D6(1,0,0,0,1,0), D6(0,0,0,0,0,1) + 2*D6(1,0,0,0,0,0)] sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8=WeylCharacterRing("E8",style="coroots",cache=True) sage: D7=WeylCharacterRing("D7",style="coroots",cache=True) sage: E8(1,0,0,0,0,0,0,0).branch(D7,rule="levi") # not tested (very long time) (160s) 3*D7(0,0,0,0,0,0,0) + 2*D7(0,0,0,0,0,1,0) + 2*D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) + 2*D7(0,0,1,0,0,0,0) + D7(0,0,0,1,0,0,0) + D7(1,0,0,0,0,1,0) + D7(1,0,0,0,0,0,1) + D7(2,0,0,0,0,0,0) sage: E8(0,0,0,0,0,0,0,1).branch(D7,rule="levi") # long time (3s) D7(0,0,0,0,0,0,0) + D7(0,0,0,0,0,1,0) + D7(0,0,0,0,0,0,1) + 2*D7(1,0,0,0,0,0,0) + D7(0,1,0,0,0,0,0) sage: [F4(fw).branch(B3,rule="levi") for fw in F4.fundamental_weights()] # long time (36s) [B3(0,0,0) + 2*B3(1/2,1/2,1/2) + 2*B3(1,0,0) + B3(1,1,0), B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 5*B3(1,0,0) + 7*B3(1,1,0) + 3*B3(1,1,1) + 6*B3(3/2,1/2,1/2) + 2*B3(3/2,3/2,1/2) + B3(2,0,0) + 2*B3(2,1,0) + B3(2,1,1), 3*B3(0,0,0) + 6*B3(1/2,1/2,1/2) + 4*B3(1,0,0) + 3*B3(1,1,0) + B3(1,1,1) + 2*B3(3/2,1/2,1/2), 3*B3(0,0,0) + 2*B3(1/2,1/2,1/2) + B3(1,0,0)] sage: [F4(fw).branch(C3,rule="levi") for fw in F4.fundamental_weights()] # long time (6s) [3*C3(0,0,0) + 2*C3(1,1,1) + C3(2,0,0), 3*C3(0,0,0) + 6*C3(1,1,1) + 4*C3(2,0,0) + 2*C3(2,1,0) + 3*C3(2,2,0) + C3(2,2,2) + C3(3,1,0) + 2*C3(3,1,1), 2*C3(1,0,0) + 3*C3(1,1,0) + C3(2,0,0) + 2*C3(2,1,0) + C3(2,1,1), 2*C3(1,0,0) + C3(1,1,0)] sage: A1xA1 = WeylCharacterRing("A1xA1") sage: [A3(hwv).branch(A1xA1,rule="levi") for hwv in A3.fundamental_weights()] [A1xA1(0,0,1,0) + A1xA1(1,0,0,0), A1xA1(0,0,1,1) + A1xA1(1,0,1,0) + A1xA1(1,1,0,0), A1xA1(1,0,1,1) + A1xA1(1,1,1,0)] sage: A1xB1=WeylCharacterRing("A1xB1",style="coroots") sage: [B3(x).branch(A1xB1,rule="levi") for x in B3.fundamental_weights()] [A1xB1(0,2) + 2*A1xB1(1,0), 3*A1xB1(0,0) + A1xB1(0,2) + 2*A1xB1(1,2) + A1xB1(2,0), 2*A1xB1(0,1) + A1xB1(1,1)] EXAMPLES: (Automorphic type, including D4 triality) :: sage: [A3(chi).branch(A3,rule="automorphic") for chi in A3.fundamental_weights()] [A3(0,0,0,-1), A3(0,0,-1,-1), A3(0,-1,-1,-1)] sage: [D4(chi).branch(D4,rule="automorphic") for chi in D4.fundamental_weights()] [D4(1,0,0,0), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1/2,1/2,1/2,-1/2)] sage: [D4(chi).branch(D4,rule="triality") for chi in D4.fundamental_weights()] [D4(1/2,1/2,1/2,-1/2), D4(1,1,0,0), D4(1/2,1/2,1/2,1/2), D4(1,0,0,0)] EXAMPLES: (Symmetric type) :: sage: [w.branch(B2,rule="symmetric") for w in [A4(1,0,0,0,0),A4(1,1,0,0,0),A4(1,1,1,0,0),A4(2,0,0,0,0)]] [B2(1,0), B2(1,1), B2(1,1), B2(0,0) + B2(2,0)] sage: [A5(w).branch(C3,rule="symmetric") for w in A5.fundamental_weights()] [C3(1,0,0), C3(0,0,0) + C3(1,1,0), C3(1,0,0) + C3(1,1,1), C3(0,0,0) + C3(1,1,0), C3(1,0,0)] sage: [A5(w).branch(D3,rule="symmetric") for w in A5.fundamental_weights()] [D3(1,0,0), D3(1,1,0), D3(1,1,-1) + D3(1,1,1), D3(1,1,0), D3(1,0,0)] sage: [D4(x).branch(B3,rule="symmetric") for x in D4.fundamental_weights()] [B3(0,0,0) + B3(1,0,0), B3(1,0,0) + B3(1,1,0), B3(1/2,1/2,1/2), B3(1/2,1/2,1/2)] sage: [D4(x).branch(G2,rule="symmetric") for x in D4.fundamental_weights()] [G2(0,0,0) + G2(1,0,-1), 2*G2(1,0,-1) + G2(2,-1,-1), G2(0,0,0) + G2(1,0,-1), G2(0,0,0) + G2(1,0,-1)] sage: [E6(fw).branch(F4,rule="symmetric") for fw in E6.fundamental_weights()] # long time (36s) [F4(0,0,0,0) + F4(0,0,0,1), F4(0,0,0,1) + F4(1,0,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(1,0,0,0) + 2*F4(0,0,1,0) + F4(1,0,0,1) + F4(0,1,0,0), F4(0,0,0,1) + F4(1,0,0,0) + F4(0,0,1,0), F4(0,0,0,0) + F4(0,0,0,1)] EXAMPLES: (Extended type) :: sage: [B3(x).branch(D3,rule="extended") for x in B3.fundamental_weights()] [D3(0,0,0) + D3(1,0,0), D3(1,0,0) + D3(1,1,0), D3(1/2,1/2,-1/2) + D3(1/2,1/2,1/2)] sage: [G2(w).branch(A2, rule="extended") for w in G2.fundamental_weights()] [A2(0,0,0) + A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3), A2(1/3,1/3,-2/3) + A2(2/3,-1/3,-1/3) + A2(1,0,-1)] sage: [F4(fw).branch(B4,rule="extended") for fw in F4.fundamental_weights()] # long time (9s) [B4(1/2,1/2,1/2,1/2) + B4(1,1,0,0), B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2) + B4(3/2,3/2,1/2,1/2) + B4(2,1,1,0), B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0) + B4(1,1,0,0) + B4(1,1,1,0) + B4(3/2,1/2,1/2,1/2), B4(0,0,0,0) + B4(1/2,1/2,1/2,1/2) + B4(1,0,0,0)] sage: E6 = WeylCharacterRing("E6", style="coroots") sage: A2xA2xA2=WeylCharacterRing("A2xA2xA2",style="coroots") sage: A5xA1=WeylCharacterRing("A5xA1",style="coroots") sage: G2 = WeylCharacterRing("G2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: F4 = WeylCharacterRing("F4",style="coroots") sage: A3xA1 = WeylCharacterRing("A3xA1", style="coroots") sage: A2xA2 = WeylCharacterRing("A2xA2", style="coroots") sage: A1xC3 = WeylCharacterRing("A1xC3",style="coroots") sage: E6(1,0,0,0,0,0).branch(A5xA1,rule="extended") # (0.7s) A5xA1(0,0,0,1,0,0) + A5xA1(1,0,0,0,0,1) sage: E6(1,0,0,0,0,0).branch(A2xA2xA2, rule="extended") # (0.7s) A2xA2xA2(0,0,0,1,1,0) + A2xA2xA2(0,1,1,0,0,0) + A2xA2xA2(1,0,0,0,0,1) sage: E7=WeylCharacterRing("E7",style="coroots") sage: A7=WeylCharacterRing("A7",style="coroots") sage: E7(1,0,0,0,0,0,0).branch(A7,rule="extended") # long time (5s) A7(0,0,0,1,0,0,0) + A7(1,0,0,0,0,0,1) sage: E8=WeylCharacterRing("E8",cache=true,style="coroots") sage: D8=WeylCharacterRing("D8",cache=true,style="coroots") sage: E8(0,0,0,0,0,0,0,1).branch(D8,rule="extended") # long time (19s) D8(0,0,0,0,0,0,1,0) + D8(0,1,0,0,0,0,0,0) sage: F4(1,0,0,0).branch(A1xC3,rule="extended") # (0.7s) A1xC3(0,2,0,0) + A1xC3(1,0,0,1) + A1xC3(2,0,0,0) sage: G2(0,1).branch(A1xA1, rule="extended") A1xA1(0,2) + A1xA1(2,0) + A1xA1(3,1) sage: F4(0,0,0,1).branch(A2xA2, rule="extended") # (0.4s) A2xA2(0,0,1,1) + A2xA2(0,1,0,1) + A2xA2(1,0,1,0) sage: F4(0,0,0,1).branch(A3xA1,rule="extended") # (0.34s) A3xA1(0,0,0,0) + A3xA1(0,0,0,2) + A3xA1(0,0,1,1) + A3xA1(0,1,0,0) + A3xA1(1,0,0,1) sage: D4=WeylCharacterRing("D4",style="coroots") sage: D2xD2=WeylCharacterRing("D2xD2",style="coroots") # We get D4 => A1xA1xA1xA1 by remembering that A1xA1 = D2. sage: [D4(fw).branch(D2xD2, rule="extended") for fw in D4.fundamental_weights()] [D2xD2(0,0,1,1) + D2xD2(1,1,0,0), D2xD2(0,0,2,0) + D2xD2(0,0,0,2) + D2xD2(2,0,0,0) + D2xD2(1,1,1,1) + D2xD2(0,2,0,0), D2xD2(1,0,0,1) + D2xD2(0,1,1,0), D2xD2(1,0,1,0) + D2xD2(0,1,0,1)] EXAMPLES: (Tensor type) :: sage: A5=WeylCharacterRing("A5", style="coroots") sage: A2xA1=WeylCharacterRing("A2xA1", style="coroots") sage: [A5(hwv).branch(A2xA1, rule="tensor") for hwv in A5.fundamental_weights()] [A2xA1(1,0,1), A2xA1(0,1,2) + A2xA1(2,0,0), A2xA1(0,0,3) + A2xA1(1,1,1), A2xA1(1,0,2) + A2xA1(0,2,0), A2xA1(0,1,1)] sage: B4=WeylCharacterRing("B4",style="coroots") sage: B1xB1=WeylCharacterRing("B1xB1",style="coroots") sage: [B4(f).branch(B1xB1,rule="tensor") for f in B4.fundamental_weights()] [B1xB1(2,2), B1xB1(0,2) + B1xB1(2,0) + B1xB1(2,4) + B1xB1(4,2), B1xB1(0,2) + B1xB1(0,6) + B1xB1(2,0) + B1xB1(2,2) + B1xB1(2,4) + B1xB1(4,2) + B1xB1(4,4) + B1xB1(6,0), B1xB1(1,3) + B1xB1(3,1)] sage: D4=WeylCharacterRing("D4",style="coroots") sage: C2xC1=WeylCharacterRing("C2xC1",style="coroots") sage: [D4(f).branch(C2xC1,rule="tensor") for f in D4.fundamental_weights()] [C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,2) + C2xC1(2,0,0), C2xC1(1,0,1), C2xC1(0,0,2) + C2xC1(0,1,0)] sage: C3=WeylCharacterRing("C3",style="coroots") sage: B1xC1=WeylCharacterRing("B1xC1",style="coroots") sage: [C3(f).branch(B1xC1,rule="tensor") for f in C3.fundamental_weights()] [B1xC1(2,1), B1xC1(2,2) + B1xC1(4,0), B1xC1(0,3) + B1xC1(4,1)] EXAMPLES: (Symmetric Power) :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: B3=WeylCharacterRing("B3",style="coroots") sage: C3=WeylCharacterRing("C3",style="coroots") sage: [B3(fw).branch(A1,rule="symmetric_power") for fw in B3.fundamental_weights()] [A1(6), A1(2) + A1(6) + A1(10), A1(0) + A1(6)] sage: [C3(fw).branch(A1,rule="symmetric_power") for fw in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] EXAMPLES: (Miscellaneous type) :: sage: G2 = WeylCharacterRing("G2") sage: [fw1, fw2, fw3] = B3.fundamental_weights() sage: B3(fw1+fw3).branch(G2, rule="miscellaneous") G2(1,0,-1) + G2(2,-1,-1) + G2(2,0,-2) EXAMPLES: (Isomorphic type) :: sage: [B2(x).branch(C2, rule="isomorphic") for x in B2.fundamental_weights()] [C2(1,1), C2(1,0)] sage: [C2(x).branch(B2, rule="isomorphic") for x in C2.fundamental_weights()] [B2(1/2,1/2), B2(1,0)] sage: [A3(x).branch(D3,rule="isomorphic") for x in A3.fundamental_weights()] [D3(1/2,1/2,1/2), D3(1,0,0), D3(1/2,1/2,-1/2)] sage: [D3(x).branch(A3,rule="isomorphic") for x in D3.fundamental_weights()] [A3(1/2,1/2,-1/2,-1/2), A3(1/4,1/4,1/4,-3/4), A3(3/4,-1/4,-1/4,-1/4)] Here A3(x,y,z,w) can be understood as a representation of SL(4). The weights x,y,z,w and x+t,y+t,z+t,w+t represent the same representation of SL(4) - though not of GL(4) - since A3(x+t,y+t,z+t,w+t) is the same as A3(x,y,z,w) tensored with `det^t`. So as a representation of SL(4), A3(1/4,1/4,1/4,-3/4) is the same as A3(1,1,1,0). The exterior square representation SL(4) -> GL(6) admits an invariant symmetric bilinear form, so is a representation SL(4) -> SO(6) that lifts to an isomorphism SL(4) -> Spin(6). Conversely, there are two isomorphisms SO(6) -> SL(4), of which we've selected one. In cases like this you might prefer style="coroots". :: sage: A3 = WeylCharacterRing("A3",style="coroots") sage: D3 = WeylCharacterRing("D3",style="coroots") sage: [D3(fw) for fw in D3.fundamental_weights()] [D3(1,0,0), D3(0,1,0), D3(0,0,1)] sage: [D3(fw).branch(A3,rule="isomorphic") for fw in D3.fundamental_weights()] [A3(0,1,0), A3(0,0,1), A3(1,0,0)] sage: D2 = WeylCharacterRing("D2", style="coroots") sage: A1xA1 = WeylCharacterRing("A1xA1", style="coroots") sage: [D2(fw).branch(A1xA1,rule="isomorphic") for fw in D2.fundamental_weights()] [A1xA1(1,0), A1xA1(0,1)] EXAMPLES: (Branching rules from plethysms) This is a general rule that includes any branching rule from types A,B,C or D as a special case. Thus it could be used in place of the above rules and would give the same results. However it is most useful when branching from G to a maximal subgroup H such that rank(H) < rank(G)-1. We consider a homomorphism H --> G where G is one of SL(r+1), SO(2r+1), Sp(2r) or SO(2r). The function branching_rule_from_plethysm produces the corresponding branching rule. The main ingredient is the character chi of the representation of H that is the homomorphism to GL(r+1), GL(2r+1) or GL(2r). This rule is so powerful that it contains the other rules implemented above as special cases. First let us consider the symmetric fifth power representation of SL(2). :: sage: A1=WeylCharacterRing("A1",style="coroots") sage: chi=A1([5]) sage: chi.degree() 6 sage: chi.frobenius_schur_indicator() -1 This confirms that the character has degree 6 and is symplectic, so it corresponds to a homomorphism SL(2) --> Sp(6), and there is a corresponding branching rule C3 => A1. :: sage: C3 = WeylCharacterRing("C3",style="coroots") sage: sym5rule = branching_rule_from_plethysm(chi,"C3") sage: [C3(hwv).branch(A1,rule=sym5rule) for hwv in C3.fundamental_weights()] [A1(5), A1(4) + A1(8), A1(3) + A1(9)] This is identical to the results we would obtain using rule="symmetric_power". The next example gives a branching not available by other standard rules. :: sage: G2 = WeylCharacterRing("G2",style="coroots") sage: D7 = WeylCharacterRing("D7",style="coroots") sage: ad=G2(0,1); ad.degree(); ad.frobenius_schur_indicator() 14 1 sage: spin = D7(0,0,0,0,0,1,0); spin.degree() 64 sage: spin.branch(G2, rule=branching_rule_from_plethysm(ad, "D7")) G2(1,1) We have confirmed that the adjoint representation of G2 gives a homomorphism into SO(14), and that the pullback of the one of the two 64 dimensional spin representations to SO(14) is an irreducible representation of G2. BRANCHING FROM A REDUCIBLE ROOT SYSTEM If you are branching from a reducible root system, the rule is a list of rules, one for each component type in the root system. The rules in the list are given in pairs [type, rule], where type is the root system to be branched to, and rule is the branching rule. :: sage: D4 = WeylCharacterRing("D4",style="coroots") sage: D2xD2 = WeylCharacterRing("D2xD2",style="coroots") sage: A1xA1xA1xA1 = WeylCharacterRing("A1xA1xA1xA1",style="coroots") sage: rr = [["A1xA1","isomorphic"],["A1xA1","isomorphic"]] sage: [D4(fw) for fw in D4.fundamental_weights()] [D4(1,0,0,0), D4(0,1,0,0), D4(0,0,1,0), D4(0,0,0,1)] sage: [D4(fw).branch(D2xD2,rule="extended").branch(A1xA1xA1xA1,rule=rr) for fw in D4.fundamental_weights()] [A1xA1xA1xA1(0,0,1,1) + A1xA1xA1xA1(1,1,0,0), A1xA1xA1xA1(0,0,0,2) + A1xA1xA1xA1(0,0,2,0) + A1xA1xA1xA1(0,2,0,0) + A1xA1xA1xA1(1,1,1,1) + A1xA1xA1xA1(2,0,0,0), A1xA1xA1xA1(0,1,1,0) + A1xA1xA1xA1(1,0,0,1), A1xA1xA1xA1(0,1,0,1) + A1xA1xA1xA1(1,0,1,0)] WRITING YOUR OWN RULES Suppose you want to branch from a group G to a subgroup H. Arrange the embedding so that a Cartan subalgebra U of H is contained in a Cartan subalgebra T of G. There is thus a mapping from the weight spaces Lie(T)* --> Lie(U)*. Two embeddings will produce identical branching rules if they differ by an element of the Weyl group of H. The RULE is this map Lie(T)* = G.space() to Lie(U)* = H.space(), which you may implement as a function. As an example, let us consider how to implement the branching rule A3 => C2. Here H = C2 = Sp(4) embedded as a subgroup in A3 = GL(4). The Cartan subalgebra U consists of diagonal matrices with eigenvalues u1, u2, -u2, -u1. The C2.space() is the two dimensional vector spaces consisting of the linear functionals u1 and u2 on U. On the other hand Lie(T) is RR^4. A convenient way to see the restriction is to think of it as the adjoint of the map [u1,u2] -> [u1,u2,-u2,-u1], that is, [x0,x1,x2,x3] -> [x0-x3,x1-x2]. Hence we may encode the rule: :: def rule(x): return [x[0]-x[3],x[1]-x[2]] or simply: :: rule = lambda x : [x[0]-x[3],x[1]-x[2]] EXAMPLES:: sage: A3 = WeylCharacterRing(['A',3]) sage: C2 = WeylCharacterRing(['C',2]) sage: rule = lambda x : [x[0]-x[3],x[1]-x[2]] sage: branch_weyl_character(A3([1,1,0,0]),A3,C2,rule) C2(0,0) + C2(1,1) sage: A3(1,1,0,0).branch(C2, rule) == C2(0,0) + C2(1,1) True """ if type(rule) == str: rule = get_branching_rule(R._cartan_type, S._cartan_type, rule) elif R._cartan_type.is_compound(): Rtypes = R._cartan_type.component_types() Stypes = [CartanType(l[0]) for l in rule] rules = [l[1] for l in rule] ntypes = len(Rtypes) rule_list = [get_branching_rule(Rtypes[i], Stypes[i], rules[i]) for i in range(ntypes)] shifts = R._cartan_type._shifts def rule(x): yl = [] for i in range(ntypes): yl.append(rule_list[i](x[shifts[i]:shifts[i+1]])) return flatten(yl) mdict = {} for k in chi._mdict: if S._style == "coroots": if S._cartan_type.is_atomic() and S._cartan_type[0] == 'E': if S._cartan_type[1] == 6: h = S._space(rule(list(k.to_vector()))) h = S.coerce_to_e6(h) elif S._cartan_type[1] == 7: h = S.coerce_to_e7(S._space(rule(list(k.to_vector())))) else: h = S.coerce_to_sl(S._space(rule(list(k.to_vector())))) else: h = S._space(rule(list(k.to_vector()))) if h in mdict: mdict[h] += chi._mdict[k] else: mdict[h] = chi._mdict[k] hdict = S.char_from_weights(mdict) return WeylCharacter(S, hdict, mdict)
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elif rule == "extended": if not s == r:
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elif rule == "extended" or rule == "orthogonal_sum": if rule == "extended" and not s == r:
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def rule(x) : x[len(x)-1] = -x[len(x)-1]; return x
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if x == 0:
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if x == 0 and not x in self._space:
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def __call__(self, *args): """ Coerces the element into the ring. INPUT: - ``x`` - a ring element EXAMPLES:: sage: a2 = WeightRing(WeylCharacterRing(['A',2])) sage: a2(-1) -a2(0,0,0) """ if len(args) == 1: x = args[0] else: x = args if x == 0: return WeightRingElement(self, {}) if x in ZZ: mdict = {self._origin: x} return WeightRingElement(self, mdict) if is_Element(x): P = x.parent() if P is self: return x elif x in self.base_ring(): mdict = {self._origin: x} return WeightRingElement(self, mdict) try: if x.parent() == self._parent: return WeightRingElement(self, x._mdict) except AttributeError: pass x = self._space(x) mdict = {x: 1} return WeightRingElement(self, mdict)
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\log(n)` time (and in linear time if all weights are equal). On the other hand, if one is given a large (possibly
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\log(n)` time (and in linear time if all weights are equal) where `n = V + E`. On the other hand, if one is given a large (possibly
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def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices.
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sage: P, = E.gens()
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sage: P = E([0,-1])
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sage: def naive_height(P):
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AUTHORS: - Robert Bradshaw (2007-10): numerical algorithm - Robert Bradshaw (2008-10): algebraic algorithm
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def minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0): r""" Return the minimal polynomial of self, if possible. INPUT: - ``var`` - polynomial variable name (default 'x') - ``algorithm`` - 'algebraic' or 'numerical' (default both, but with numerical first) - ``bits`` - the number of bits to use in numerical approx - ``degree`` - the expected algebraic degree - ``epsilon`` - return without error as long as f(self) epsilon, in the case that the result cannot be proven. All of the above parameters are optional, with epsilon=0, bits and degree tested up to 1000 and 24 by default respectively. The numerical algorithm will be faster if bits and/or degree are given explicitly. The algebraic algorithm ignores the last three parameters. OUTPUT: The minimal polynomial of self. If the numerical algorithm is used then it is proved symbolically when epsilon=0 (default). If the minimal polynomial could not be found, two distinct kinds of errors are raised. If no reasonable candidate was found with the given bit/degree parameters, a ``ValueError`` will be raised. If a reasonable candidate was found but (perhaps due to limits in the underlying symbolic package) was unable to be proved correct, a ``NotImplementedError`` will be raised. ALGORITHM: Two distinct algorithms are used, depending on the algorithm parameter. By default, the numerical algorithm is attempted first, then the algebraic one. Algebraic: Attempt to evaluate this expression in QQbar, using cyclotomic fields to resolve exponential and trig functions at rational multiples of pi, field extensions to handle roots and rational exponents, and computing compositums to represent the full expression as an element of a number field where the minimal polynomial can be computed exactly. The bits, degree, and epsilon parameters are ignored. Numerical: Computes a numerical approximation of ``self`` and use PARI's algdep to get a candidate minpoly `f`. If `f(\mathtt{self})`, evaluated to a higher precision, is close enough to 0 then evaluate `f(\mathtt{self})` symbolically, attempting to prove vanishing. If this fails, and ``epsilon`` is non-zero, return `f` if and only if `f(\mathtt{self}) < \mathtt{epsilon}`. Otherwise raise a ``ValueError`` (if no suitable candidate was found) or a ``NotImplementedError`` (if a likely candidate was found but could not be proved correct). EXAMPLES: First some simple examples:: sage: sqrt(2).minpoly() x^2 - 2 sage: minpoly(2^(1/3)) x^3 - 2 sage: minpoly(sqrt(2) + sqrt(-1)) x^4 - 2*x^2 + 9 sage: minpoly(sqrt(2)-3^(1/3)) x^6 - 6*x^4 + 6*x^3 + 12*x^2 + 36*x + 1 Works with trig and exponential functions too. :: sage: sin(pi/3).minpoly() x^2 - 3/4 sage: sin(pi/7).minpoly() x^6 - 7/4*x^4 + 7/8*x^2 - 7/64 sage: minpoly(exp(I*pi/17)) x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Here we verify it gives the same result as the abstract number field. :: sage: (sqrt(2) + sqrt(3) + sqrt(6)).minpoly() x^4 - 22*x^2 - 48*x - 23 sage: K.<a,b> = NumberField([x^2-2, x^2-3]) sage: (a+b+a*b).absolute_minpoly() x^4 - 22*x^2 - 48*x - 23 The minpoly function is used implicitly when creating number fields:: sage: x = var('x') sage: eqn = x^3 + sqrt(2)*x + 5 == 0 sage: a = solve(eqn, x)[0].rhs() sage: QQ[a] Number Field in a with defining polynomial x^6 + 10*x^3 - 2*x^2 + 25 Here we solve a cubic and then recover it from its complicated radical expansion. :: sage: f = x^3 - x + 1 sage: a = f.solve(x)[0].rhs(); a -1/2*(I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(23) - 1/2)^(1/3) - 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(3)*sqrt(23) - 1/2)^(1/3) sage: a.minpoly() x^3 - x + 1 Note that simplification may be necessary to see that the minimal polynomial is correct. :: sage: a = sqrt(2)+sqrt(3)+sqrt(5) sage: f = a.minpoly(); f x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576 sage: f(a) ((((sqrt(2) + sqrt(3) + sqrt(5))^2 - 40)*(sqrt(2) + sqrt(3) + sqrt(5))^2 + 352)*(sqrt(2) + sqrt(3) + sqrt(5))^2 - 960)*(sqrt(2) + sqrt(3) + sqrt(5))^2 + 576 sage: f(a).expand() 0 Here we show use of the ``epsilon`` parameter. That this result is actually exact can be shown using the addition formula for sin, but maxima is unable to see that. :: sage: a = sin(pi/5) sage: a.minpoly(algorithm='numerical') Traceback (most recent call last): ... NotImplementedError: Could not prove minimal polynomial x^4 - 5/4*x^2 + 5/16 (epsilon 0.00000000000000e-1) sage: f = a.minpoly(algorithm='numerical', epsilon=1e-100); f x^4 - 5/4*x^2 + 5/16 sage: f(a).numerical_approx(100) 0.00000000000000000000000000000 The degree must be high enough (default tops out at 24). :: sage: a = sqrt(3) + sqrt(2) sage: a.minpoly(algorithm='numerical', bits=100, degree=3) Traceback (most recent call last): ... ValueError: Could not find minimal polynomial (100 bits, degree 3). sage: a.minpoly(algorithm='numerical', bits=100, degree=10) x^4 - 10*x^2 + 1 There is a difference between algorithm='algebraic' and algorithm='numerical':: sage: cos(pi/33).minpoly(algorithm='algebraic') x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024 sage: cos(pi/33).minpoly(algorithm='numerical') Traceback (most recent call last): ... NotImplementedError: Could not prove minimal polynomial x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024 (epsilon ...) Sometimes it fails, as it must given that some numbers aren't algebraic:: sage: sin(1).minpoly(algorithm='numerical') Traceback (most recent call last): ... ValueError: Could not find minimal polynomial (1000 bits, degree 24). .. note:: Of course, failure to produce a minimal polynomial does not necessarily indicate that this number is transcendental. AUTHORS: - Robert Bradshaw (2007-10): numerical algorithm - Robert Bradshaw (2008-10): algebraic algorithm """ if algorithm is None or algorithm.startswith('numeric'): bits_list = [bits] if bits else [100,200,500,1000] degree_list = [degree] if degree else [2,4,8,12,24] for bits in bits_list: a = ex.numerical_approx(bits) check_bits = int(1.25 * bits + 80) aa = ex.numerical_approx(check_bits) for degree in degree_list: f = QQ[var](algdep(a, degree)) # TODO: use the known_bits parameter? # If indeed we have found a minimal polynomial, # it should be accurate to a much higher precision. error = abs(f(aa)) dx = ~RR(Integer(1) << (check_bits - degree - 2)) expected_error = abs(f.derivative()(CC(aa))) * dx if error < expected_error: # Degree might have been an over-estimate, factor because we want (irreducible) minpoly. ff = f.factor() for g, e in ff: lead = g.leading_coefficient() if lead != 1: g = g / lead expected_error = abs(g.derivative()(CC(aa))) * dx error = abs(g(aa)) if error < expected_error: # See if we can prove equality exactly if g(ex).simplify_trig().simplify_radical() == 0: return g # Otherwise fall back to numerical guess elif epsilon and error < epsilon: return g elif algorithm is not None: raise NotImplementedError, "Could not prove minimal polynomial %s (epsilon %s)" % (g, RR(error).str(no_sci=False)) if algorithm is not None: raise ValueError, "Could not find minimal polynomial (%s bits, degree %s)." % (bits, degree) if algorithm is None or algorithm == 'algebraic': from sage.rings.all import QQbar return QQ[var](QQbar(ex).minpoly()) raise ValueError, "Unknown algorithm: %s" % algorithm
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AUTHORS: - Golam Mortuza Hossain (2009-06-15)
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def _limit_latex_(self, f, x, a): r""" Return latex expression for limit of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _limit_latex_ sage: var('x,a') (x, a) sage: f = function('f',x) sage: _limit_latex_(0, f, x, a) '\\lim_{x \\to a}\\, f\\left(x\\right)' sage: latex(limit(f, x=oo)) \lim_{x \to +\infty}\, f\left(x\right) AUTHORS: - Golam Mortuza Hossain (2009-06-15) """ return "\\lim_{%s \\to %s}\\, %s"%(latex(x), latex(a), latex(f))
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|
AUTHORS: - Golam Mortuza Hossain (2009-06-22)
|
def _laplace_latex_(self, *args): r""" Return LaTeX expression for Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _laplace_latex_ sage: var('s,t') (s, t) sage: f = function('f',t) sage: _laplace_latex_(0,f,t,s) '\\mathcal{L}\\left(f\\left(t\\right), t, s\\right)' sage: latex(laplace(f, t, s)) \mathcal{L}\left(f\left(t\right), t, s\right) AUTHORS: - Golam Mortuza Hossain (2009-06-22) """ return "\\mathcal{L}\\left(%s\\right)"%(', '.join([latex(x) for x in args]))
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|
AUTHORS: - Golam Mortuza Hossain (2009-06-22)
|
def _inverse_laplace_latex_(self, *args): r""" Return LaTeX expression for inverse Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _inverse_laplace_latex_ sage: var('s,t') (s, t) sage: F = function('F',s) sage: _inverse_laplace_latex_(0,F,s,t) '\\mathcal{L}^{-1}\\left(F\\left(s\\right), s, t\\right)' sage: latex(inverse_laplace(F,s,t)) \mathcal{L}^{-1}\left(F\left(s\right), s, t\right) AUTHORS: - Golam Mortuza Hossain (2009-06-22) """ return "\\mathcal{L}^{-1}\\left(%s\\right)"%(', '.join([latex(x) for x in args]))
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|
sage: c = c2 = 1
|
sage: c == c2 calling __eq__ defined in Metaclass True
|
def metaclass(name, bases): """ Creates a new class in this metaclass INPUT:: - name: a string - bases: a tuple of classes EXAMPLES:: sage: from sage.misc.test_class_pickling import metaclass, bar sage: c = metaclass("foo2", (object, bar,)) constructing class sage: c <class 'sage.misc.test_class_pickling.foo2'> sage: type(c) <class 'sage.misc.test_class_pickling.Metaclass'> sage: c.__bases__ (<type 'object'>, <class sage.misc.test_class_pickling.bar at ...>) """ print "constructing class" result = Metaclass(name, bases, dict()) result.reduce_args = (name, bases) return result
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if order is Infinity:
|
if order == 1: if isinstance(w, (tuple,str,list)): length = 'finite' elif isinstance(w, FiniteWord_class): length = sum(self._morph[a].length() * b for (a,b) in w.evaluation_dict().iteritems()) elif hasattr(w, '__iter__'): length = Infinity datatype = 'iter' elif w in self._domain.alphabet(): return self._morph[w] else: raise TypeError, "Don't know how to handle an input (=%s) that is not iterable or not in the domain alphabet."%w return self.codomain()((x for y in w for x in self._morph[y]), length=length, datatype=datatype) elif order is Infinity:
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def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT:
|
if not isinstance(order, (int,Integer)) or order < 0 : raise TypeError, "order (%s) must be a positive integer or plus Infinity" % order
|
elif isinstance(order, (int,Integer)) and order > 1: return self(self(w, order-1),datatype=datatype)
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def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT:
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elif order == 1: if isinstance(w, (tuple,str,list)): length = 'finite' elif isinstance(w, FiniteWord_class): length = sum(self._morph[a].length() * b for (a,b) in w.evaluation_dict().iteritems()) elif hasattr(w, '__iter__'): length = Infinity datatype = 'iter' elif w in self._domain.alphabet(): w = [w] length = 'finite' else: raise TypeError, "Don't know how to handle an input (=%s) that is not iterable or not in the domain alphabet."%w return self.codomain()((x for y in w for x in self._morph[y]), length=length, datatype=datatype) elif order > 1: return self(self(w, order-1),datatype=datatype)
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else: raise TypeError, "order (%s) must be a positive integer or plus Infinity" % order
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def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT:
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image = self(letter)
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image = self.image(letter)
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def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT:
|
def letter_iterator(self, letter):
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def _fixed_point_iterator(self, letter):
|
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``.
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sage: list(m.letter_iterator('b')) Traceback (most recent call last): ... TypeError: self must be prolongable on b sage: list(m.letter_iterator('a'))
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sage: list(m._fixed_point_iterator('a'))
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def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``.
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sage: m = WordMorphism('a->aa,b->aac') sage: list(m.letter_iterator('a')) Traceback (most recent call last): ... TypeError: self (=WordMorphism: a->aa, b->aac) is not a endomorphism """ if not self.is_endomorphism(): raise TypeError, "self (=%s) is not a endomorphism"%self if not self.is_prolongable(letter=letter): raise TypeError, "self must be prolongable on %s"%letter w = list(self(letter))
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The morphism must be prolongable on the letter:: sage: list(m._fixed_point_iterator('b')) Traceback (most recent call last): ... IndexError: pop from empty list The morphism must be an endomorphism:: sage: m = WordMorphism('a->ac,b->aac') sage: list(m._fixed_point_iterator('a')) Traceback (most recent call last): ... KeyError: 'c' We check that sage: s = WordMorphism({('a', 1):[('a', 1), ('a', 2)], ('a', 2):[('a', 1)]}) sage: it = s._fixed_point_iterator(('a',1)) sage: it.next() ('a', 1) """ w = list(self.image(letter))
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def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``.
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for a in self(w.pop(0)):
|
for a in self.image(w.pop(0)):
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def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``.
|
w.extend(self(w[0]))
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w.extend(self.image(w[0]))
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def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``.
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image = self(letter)
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image = self.image(letter)
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def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``.
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w = self.codomain()(self.letter_iterator(letter), datatype='iter')
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w = self.codomain()(self._fixed_point_iterator(letter), datatype='iter')
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def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``.
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if os.uname()[0][:6] == 'CYGWIN':
|
if os.uname()[0][:6] == 'CYGWIN' and package is not None:
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def install_package(package=None, force=False): """ Install a package or return a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ``package`` - optional; if specified, install the given package. If not, list all installed packages. IMPLEMENTATION: calls 'sage -f'. .. seealso:: :func:`optional_packages`, :func:`upgrade` """ global __installed_packages if os.uname()[0][:6] == 'CYGWIN': print "install_package may not work correctly under Microsoft Windows" print "since you can't change an opened file. Quit all" print "instances of sage and use 'sage -i' instead or" print "use the force option to install_package." return if package is None: if __installed_packages is None: X = os.popen('sage -f').read().split('\n') i = X.index('Currently installed packages:') X = [Y for Y in X[i+1:] if Y != ''] X.sort() __installed_packages = X return __installed_packages # Get full package name if force: S = [P for P in standard_packages()[0] if P.startswith(package)] O = [P for P in optional_packages()[0] if P.startswith(package)] E = [P for P in experimental_packages()[0] if P.startswith(package)] else: S,O,E = [], [], [] S.extend([P for P in standard_packages()[1] if P.startswith(package)]) O.extend([P for P in optional_packages()[1] if P.startswith(package)]) E.extend([P for P in experimental_packages()[1] if P.startswith(package)]) L = S+O+E if len(L)>1: if force: print "Possible package names starting with '%s' are:"%(package) else: print "Possible names of non-installed packages starting with '%s':"%(package) for P in L: print " ", P raise ValueError, "There is more than one package name starting with '%s'. Please specify!"%(package) if len(L)==0: if not force: if is_package_installed(package): raise ValueError, "Package is already installed. Try install_package('%s',force=True)"%(package) raise ValueError, "There is no package name starting with '%s'."%(package) os.system('sage -f "%s"'%(L[0])) __installed_packages = None
|
return
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def install_package(package=None, force=False): """ Install a package or return a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ``package`` - optional; if specified, install the given package. If not, list all installed packages. IMPLEMENTATION: calls 'sage -f'. .. seealso:: :func:`optional_packages`, :func:`upgrade` """ global __installed_packages if os.uname()[0][:6] == 'CYGWIN': print "install_package may not work correctly under Microsoft Windows" print "since you can't change an opened file. Quit all" print "instances of sage and use 'sage -i' instead or" print "use the force option to install_package." return if package is None: if __installed_packages is None: X = os.popen('sage -f').read().split('\n') i = X.index('Currently installed packages:') X = [Y for Y in X[i+1:] if Y != ''] X.sort() __installed_packages = X return __installed_packages # Get full package name if force: S = [P for P in standard_packages()[0] if P.startswith(package)] O = [P for P in optional_packages()[0] if P.startswith(package)] E = [P for P in experimental_packages()[0] if P.startswith(package)] else: S,O,E = [], [], [] S.extend([P for P in standard_packages()[1] if P.startswith(package)]) O.extend([P for P in optional_packages()[1] if P.startswith(package)]) E.extend([P for P in experimental_packages()[1] if P.startswith(package)]) L = S+O+E if len(L)>1: if force: print "Possible package names starting with '%s' are:"%(package) else: print "Possible names of non-installed packages starting with '%s':"%(package) for P in L: print " ", P raise ValueError, "There is more than one package name starting with '%s'. Please specify!"%(package) if len(L)==0: if not force: if is_package_installed(package): raise ValueError, "Package is already installed. Try install_package('%s',force=True)"%(package) raise ValueError, "There is no package name starting with '%s'."%(package) os.system('sage -f "%s"'%(L[0])) __installed_packages = None
|
|
return
|
return []
|
def upgrade(): """ Download and build the latest version of Sage. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. This upgrades to the latest version of core packages (optional packages are not automatically upgraded). This will not work on systems that don't have a C compiler. .. seealso:: :func:`install_package`, :func:`optional_packages` """ global __installed_packages if os.uname()[0][:6] == 'CYGWIN': print "Upgrade may not work correctly under Microsoft Windows" print "since you can't change an opened file. Quit all" print "instances of Sage and use 'sage -upgrade' instead." return os.system('sage -upgrade') __installed_packages = None print "You should quit and restart Sage now."
|
return HeckeModuleElement.__mul__(self, other)
|
return element.HeckeModuleElement.__mul__(self, other)
|
def __mul__(self, other): r""" Calculate the product self * other.
|
self._read_faces(self.poly_x("i"))
|
self._read_faces(self.poly_x("i", reduce_dimension=True))
|
def _compute_faces(self): r""" Compute and cache faces of this polytope. If this polytope is reflexive and the polar polytope was already computed, computes faces of both in order to save time and preserve the one-to-one correspondence between the faces of this polytope of dimension d and the faces of the polar polytope of codimension d+1. TESTS:: sage: o = lattice_polytope.octahedron(3) sage: v = o.__dict__.pop("_faces", None) # faces may be cached already sage: o.__dict__.has_key("_faces") False sage: o._compute_faces() sage: o.__dict__.has_key("_faces") True """ if hasattr(self, "_constructed_as_polar"): # "Polar of polar polytope" computed by poly.x may have the # order of vertices different from the original polytope. Thus, # in order to have consistent enumeration of vertices and faces # we must run poly.x on the original polytope. self._copy_faces(self._polar, reverse=True) elif hasattr(self, "_constructed_as_affine_transform"): self._copy_faces(self._original) else: self._read_faces(self.poly_x("i"))
|
Return the sequence of faces of this polytope.
|
Return the sequence of proper faces of this polytope.
|
def faces(self, dim=None, codim=None): r""" Return the sequence of faces of this polytope. If ``dim`` or ``codim`` are specified, returns a sequence of faces of the corresponding dimension or codimension. Otherwise returns the sequence of such sequences for all dimensions. EXAMPLES: All faces of the 3-dimensional octahedron:: sage: o = lattice_polytope.octahedron(3) sage: o.faces() [ [[0], [1], [2], [3], [4], [5]], [[1, 5], [0, 5], [0, 1], [3, 5], [1, 3], [4, 5], [0, 4], [3, 4], [1, 2], [0, 2], [2, 3], [2, 4]], [[0, 1, 5], [1, 3, 5], [0, 4, 5], [3, 4, 5], [0, 1, 2], [1, 2, 3], [0, 2, 4], [2, 3, 4]] ] Its faces of dimension one (i.e., edges):: sage: o.faces(dim=1) [[1, 5], [0, 5], [0, 1], [3, 5], [1, 3], [4, 5], [0, 4], [3, 4], [1, 2], [0, 2], [2, 3], [2, 4]] Its faces of codimension two (also edges):: sage: o.faces(codim=2) [[1, 5], [0, 5], [0, 1], [3, 5], [1, 3], [4, 5], [0, 4], [3, 4], [1, 2], [0, 2], [2, 3], [2, 4]] It is an error to specify both dimension and codimension at the same time, even if they do agree:: sage: o.faces(dim=1, codim=2) Traceback (most recent call last): ... ValueError: Both dim and codim are given! """ try: if dim == None and codim == None: return self._faces elif dim != None and codim == None: return self._faces[dim] elif dim == None and codim != None: return self._faces[self.dim()-codim] else: raise ValueError, "Both dim and codim are given!" except AttributeError: self._compute_faces() return self.faces(dim, codim)
|
self._points = self._embed(read_palp_matrix( self.poly_x("p", reduce_dimension=True))) self._points.set_immutable()
|
if self.dim() == 0: self._points = self._vertices else: self._points = self._embed(read_palp_matrix( self.poly_x("p", reduce_dimension=True))) self._points.set_immutable()
|
def points(self): r""" Return all lattice points of this polytope as columns of a matrix. EXAMPLES: The lattice points of the 3-dimensional octahedron and its polar cube:: sage: o = lattice_polytope.octahedron(3) sage: o.points() [ 1 0 0 -1 0 0 0] [ 0 1 0 0 -1 0 0] [ 0 0 1 0 0 -1 0] sage: cube = o.polar() sage: cube.points() [-1 1 -1 1 -1 1 -1 1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 1 1] [-1 -1 1 1 -1 -1 1 1 -1 0 0 0 1 -1 -1 -1 0 0 0 1 1 1 -1 0 0 0 1] [ 1 1 1 1 -1 -1 -1 -1 0 -1 0 1 0 -1 0 1 -1 0 1 -1 0 1 0 -1 0 1 0] Lattice points of a 2-dimensional diamond in a 3-dimensional space:: sage: m = matrix(ZZ, [[1, 0, -1, 0], ... [0, 1, 0, -1], ... [0, 0, 0, 0]]) ... sage: p = LatticePolytope(m) sage: p.points() [ 1 0 -1 0 0] [ 0 1 0 -1 0] [ 0 0 0 0 0] """ if not hasattr(self, "_points"): self._points = self._embed(read_palp_matrix( self.poly_x("p", reduce_dimension=True))) self._points.set_immutable() return self._points
|
"""
|
r"""
|
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
|
cospectral graphs.
|
cospectral graphs (lists of cadinality 1 being omitted).
|
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
|
is enough to check the spectrum of the matrix `D^{-1}A`, where D
|
is enough to check the spectrum of the matrix `D^{-1}A`, where `D`
|
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
|
if P.eval("%s %s %s"%(self.name(), P._equality_symbol(), other.name())) == P._true_symbol(): return 0 elif P.eval("%s %s %s"%(self.name(), P._lessthan_symbol(), other.name())) == P._true_symbol(): return -1 elif P.eval("%s %s %s"%(self.name(), P._greaterthan_symbol(), other.name())) == P._true_symbol(): return 1
|
try: if P.eval("%s %s %s"%(self.name(), P._equality_symbol(), other.name())) == P._true_symbol(): return 0 except RuntimeError: pass try: if P.eval("%s %s %s"%(self.name(), P._lessthan_symbol(), other.name())) == P._true_symbol(): return -1 except RuntimeError: pass try: if P.eval("%s %s %s"%(self.name(), P._greaterthan_symbol(), other.name())) == P._true_symbol(): return 1 except: pass
|
def __cmp__(self, other): P = self.parent() if P.eval("%s %s %s"%(self.name(), P._equality_symbol(), other.name())) == P._true_symbol(): return 0 elif P.eval("%s %s %s"%(self.name(), P._lessthan_symbol(), other.name())) == P._true_symbol(): return -1 elif P.eval("%s %s %s"%(self.name(), P._greaterthan_symbol(), other.name())) == P._true_symbol(): return 1
|
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