Generalized Verma module

In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries.
Definition.
Let formula_1 be a semisimple Lie algebra and formula_2 a parabolic subalgebra of formula_1. For any irreducible finite-dimensional representation formula_4 of formula_2 we define the generalized Verma module to be the relative tensor product
The action of formula_1 is left multiplication in formula_8.
If λ is the highest weight of V, we sometimes denote the Verma module by formula_9.
Note that formula_9 makes sense only for formula_2-dominant and formula_2-integral weights (see weight) formula_13.
It is well known that a parabolic subalgebra formula_2 of formula_1 determines a unique grading formula_16 so that formula_17.
Let formula_18.
It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a formula_19-module and as a formula_20-module),
In further text, we will denote a generalized Verma module simply by GVM.
Properties of GVMs.
GVM's are highest weight modules and their highest weight λ is the highest weight of the representation V. If formula_22 is the highest weight vector in V, then formula_23 is the highest weight vector in formula_9.
GVM's are weight modules, i.e. they are direct sum of its weight spaces and these weight spaces are finite-dimensional.
As all highest weight modules, GVM's are quotients of Verma modules. The kernel of the projection formula_25 is
where formula_27 is the set of those simple roots α such that the negative root spaces of root formula_28 are in formula_2 (the set S determines uniquely the subalgebra formula_2), formula_31 is the root reflection with respect to the root α and
formula_32 is the affine action of formula_31 on λ. It follows from the theory of (true) Verma modules that formula_34 is isomorphic to a unique submodule of formula_35. In (1), we identified formula_36. The sum in (1) is not direct.
In the special case when formula_37, the parabolic subalgebra formula_2 is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when formula_39, formula_40 and the GVM is isomorphic to the inducing representation V.
The GVM formula_9 is called "regular", if its highest weight λ is on the affine Weyl orbit of a dominant weight formula_42. In other word, there exist an element w of the Weyl group W such that
where formula_44 is the affine action of the Weyl group.
The Verma module formula_35 is called "singular", if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight formula_42 so that formula_47 is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
Homomorphisms of GVMs.
By a homomorphism of GVMs we mean formula_1-homomorphism.
For any two weights formula_49 a homomorphism
may exist only if formula_51 and formula_13 are linked with an affine action of the Weyl group formula_53 of the Lie algebra formula_1. This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
Unlike in the case of (true) Verma modules, the homomorphisms of GVM's are in general not injective and the dimension
may be larger than one in some specific cases.
If formula_56 is a homomorphism of (true) Verma modules, formula_57 resp. formula_58 is the kernels of the projection formula_59, resp. formula_25, then there exists a homomorphism formula_61 and f factors to a homomorphism of generalized Verma modules formula_62. Such a homomorphism (that is a factor of a homomorphism of Verma modules) is called standard. However, the standard homomorphism may be zero in some cases.
Standard.
Let us suppose that there exists a nontrivial homomorphism of true Verma moduls formula_63.
Let formula_27 be the set of those simple roots α such that the negative root spaces of root formula_28 are in formula_2 (like in section Properties).
The following theorem is proved by Lepowsky:
The standard homomorphism formula_62 is zero if and only if there exists formula_68 such that formula_69 is isomorphic to a submodule of formula_70 (formula_31 is the corresponding root reflection and formula_44 is the affine action).
The structure of GVMs on the affine orbit of a formula_1-dominant and formula_1-integral weight formula_42 can be described explicitly. If W is the Weyl group of formula_1, there exists a subset formula_77 of such elements, so that formula_78 is formula_2-dominant. It can be shown that formula_80 where formula_81 is the Weyl group of formula_2 (in particular, formula_83 does not depend on the choice of formula_42). The map formula_85 is a bijection between formula_83 and the set of GVM's with highest weights on the affine orbit of formula_42. Let as suppose that formula_88, formula_43 and formula_90 in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules formula_91 and the standard homomorphism does not make sense, see Homomorphisms of Verma modules).
The following statements follow from the above theorem and the structure of formula_83:
"Theorem." If formula_93 for some positive root formula_94 and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism formula_62.
"Theorem". The standard homomorphism formula_62 is zero if and only if there exists formula_97 such that formula_98 and formula_99.
However, if formula_42 is only dominant but not integral, there may still exist formula_2-dominant and formula_2-integral weights on its affine orbit.
The situation is even more complicated if the GVM's have singular character, i.e. there formula_51 and formula_13 are on the affine orbit of some formula_42 such that formula_47 is on the wall of the fundamental Weyl chamber.
Nonstandard.
A homomorphism formula_62 is called nonstandard, if it is not standard. It may happen that the standard homomorphism of GVMs is zero but there still exists a nonstandard homomorphism.


Grace Marie Bareis

Grace Marie Bareis (December 19, 1875 – June 15, 1962) was an American mathematician and educator who became the first person to receive a doctorate degree in mathematics from Ohio State University. Bareis was an assistant professor at Ohio State University where she taught for 40 years until her eventual retirement in 1946.
Early life and education.
Grace Marie Bareis was born on December 19, 1875 in Canal Winchester, Ohio to George Frederick Bareis and Amanda Jane. Bareis attended public school in her hometown, and then went on to attend Heidelberg University in Ohio. She graduated as valedictorian of her class and received her A.B degree in 1897. Bareis then completed part of her graduate work at Bryn Mawr College during 1897–1899, and later at Columbia University in that same year She took a brief break from school to work as a teacher at Miss Roney's School in Philadelphia where she would continue to work for the next six years. Two years after leaving Columbia University, she returned to Bryn Mawr College to continue her work on a mathematics degree. In 1906, Bareis enrolled in the graduate program at Ohio State University and after three years received the first Ph.D in mathematics from the university. Her dissertation, ""Imprimitive Substitution Groups of Degree Sixteen"," was directed by Harry W. Kuhn.
Career.
During her graduate work at Ohio State University in 1908, Bareis became an assistant professor of mathematics. She held the position until her retirement in 1946. While serving as an assistant professor, she helped to direct master thesis of her students and also held various other positions in the Education and Women College. Bareis taught mathematics to World War II veterans in a class called the "Army Specialized Training Program" and even did so two years after her retirement because of a shortage of math instructors.
Personal life.
Bareis was involved with various organizations. Her religious associations include the Evangelical and Reformed Church, Missionary Society, and Indianola Methodist Church. She was a founding member of the Mathematical Association of America, volunteered at the American Red Cross, and belonged to the Daughters of the American Revolution as well as the Ohio Historical Society. Bareis was a part of numerous groups at Ohio State University such as the OSU Faculty Club and OSU Alumnae Association. She was also a part of the Board of Trustees of Heidelberg College who awarded her an honorary doctorate in 1950.
She enjoyed gardening and tending to her farm in Brice, Ohio.
Later life and legacy.
In 1949, Bareis donated $2000 to Ohio State University to celebrate its 75th anniversary and to provide funding for a written competition between sophomores. "The Grace M. Bareis Mathematical Prize" (part of the Rasor-Bareis-Gordon) competition has been held annually in her honor ever since.
Bareis had a home in Canal Winchester, Ohio, but she began living in Columbus with her friend and colleague, Margaret F. Jones, near the start of her retirement. She lived with her for 26 years until a prolonged illness caused her death on June 15, 1962.
The Bareis Hall of Science in Heidelberg University was built in her honor.


List of things named after Niels Henrik Abel

This is the list of things named after Niels Henrik Abel (1802–1829), a Norwegian mathematician.


Herbrand quotient

In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
Definition.
If "G" is a finite cyclic group acting on a "G"-module "A", then the cohomology groups "H""n"("G","A") have period 2 for "n"≥1; in other words
an isomorphism induced by cup product with a generator of "H""2"("G",Z). (If instead we use the Tate cohomology groups then the periodicity extends down to "n"=0.)
A Herbrand module is an "A" for which the cohomology groups are finite. In this case, the Herbrand quotient "h"("G","A") is defined to be the quotient
of the order of the even and odd cohomology groups.
Alternative definition.
The quotient may be defined for a pair of endomorphisms of an Abelian group, "f" and "g", which satisfy the condition "fg" = "gf" = 0. Their Herbrand quotient "q"("f","g") is defined as
if the two indices are finite. If "G" is a cyclic group with generator γ acting on an Abelian group "A", then we recover the previous definition by taking "f" = 1 - γ and "g" = 1 + γ + γ2 + ... .
Properties.
is exact, and any two of the quotients are defined, then so is the third and
These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.


Blowing up

In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion.
Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups.
Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by blowing up singularities until they become smooth. A consequence of this is that blowups can be used to resolve the singularities of birational maps.
Classically, blowups were defined extrinsically, by first defining the blowup on spaces such as projective space using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. This is reflected in some of the terminology, such as the classical term "monoidal transformation". Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor.
A blowup can also be called "monoidal transformation", "locally quadratic transformation", "dilatation", σ-"process", or "Hopf map".
The blowup of a point in a plane.
The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example.
The blowup has a synthetic description as an incidence correspondence. Recall that the Grassmannian G(1,2) parametrizes the set of all lines through a point in the plane. The blowup of the projective plane P2 at the point "P", which we will denote "X", is
Here "Q" denotes another point and formula_2 is an element of the Grassmannian. "X" is a projective variety because it is a closed subvariety of a product of projective varieties. It comes with a natural morphism π to P2 that takes the pair formula_3 to "Q". This morphism is an isomorphism on the open subset of all points formula_3 with "Q" ≠ "P" because the line formula_2 is determined by those two points. When "Q" = "P", however, the line formula_2 can be any line through "P". These lines correspond to the space of directions through "P", which is isomorphic to P1. This P1 is called the "exceptional divisor", and by definition it is the projectivized normal space at "P". Because "P" is a point, the normal space is the same as the tangent space, so the exceptional divisor is isomorphic to the projectivized tangent space at "P".
To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P2 homogeneous coordinates ["X"0:"X"1:"X"2] in which "P" is the point ["P"0:"P"1:"P"2]. By projective duality, G(1,2) is isomorphic to P2, so we may give it homogeneous coordinates ["L"0:"L"1:"L"2]. A line formula_7 is the set of all ["X"0:"X"1:"X"2] such that "X"0"L"0 + "X"1"L"1 + "X"2"L"2 = 0. Therefore, the blowup can be described as
The blowup is an isomorphism away from "P", and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation, we may assume that "P" = [0:0:1]. Write "x" and "y" for the coordinates on the affine plane "X"2≠0. The condition "P" ∈ formula_2 implies that "L"2 = 0, so we may replace the Grassmannian with a P1. Then the blowup is the variety
It is more common to change coordinates so as to reverse one of the signs. Then the blowup can be written as
This equation is easier to generalize than the previous one.
The blowup can be easily visualized if we remove the infinity point of the Grassmannian, e.g. by setting "w" = 1, and obtain the standard saddle surface "y" = "xz" in 3D space.
The blowup can also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane A2. The normal space to the origin is the vector space "m"/"m"2, where "m" = ("x", "y") is the maximal ideal of the origin. Algebraically, the projectivization of this vector space is Proj of its symmetric algebra, that is,
In this example, this has a concrete description as
where "x" and "y" have degree 0 and "z" and "w" have degree 1.
Over the real or complex numbers, the blowup has a topological description as the connected sum formula_14. Assume that "P" is the origin in A2 ⊆ P2, and write "L" for the line at infinity. A2 \ {0} has an inversion map "t" which sends ("x", "y") to ("x"/(|"x"|2 + |"y"|2), "y"/(|"x"|2 + |"y"|2)). "t" is the circle inversion with respect to the unit sphere "S": It fixes "S", preserves each line through the origin, and exchanges the inside of the sphere with the outside. "t" extends to a continuous map P2 \ {0} → A2 by sending the line at infinity to the origin. This extension, which we also denote "t", can be used to construct the blowup. Let "C" denote the complement of the unit ball. The blowup "X" is the manifold obtained by attaching two copies of "C" along "S". "X" comes with a map π to P2 which is the identity on the first copy of "C" and "t" on the second copy of "C". This map is an isomorphism away from "P", and the fiber over "P" is the line at infinity in the second copy of "C". Each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin.
For CP2 this process ought to produce an oriented manifold. In order to make this happen, the two copies of "C" should be given opposite orientations. In symbols, "X" is formula_15, where formula_16 is CP2 with the opposite of the standard orientation.
Blowing up points in complex space.
Let "Z" be the origin in "n"-dimensional complex space, C"n". That is, "Z" is the point where the "n" coordinate functions formula_17 simultaneously vanish. Let P"n" - 1 be ("n" - 1)-dimensional complex projective space with homogeneous coordinates formula_18. Let formula_19 be the subset of C"n" × P"n" - 1 that satisfies simultaneously the equations formula_20 for "i, j" = 1, ..., "n". The projection
naturally induces a holomorphic map
This map π (or, often, the space formula_19) is called the blow-up (variously spelled blow up or blowup) of C"n".
The exceptional divisor "E" is defined as the inverse image of the blow-up locus "Z" under π. It is easy to see that
is a copy of projective space. It is an effective divisor. Away from "E", π is an isomorphism between formula_25 and C"n" \ "Z"; it is a birational map between formula_19 and C"n".
If instead we consider the holomorphic projection
we obtain the tautological line bundle of formula_28 and we can identify the exceptional divisor formula_29 with its zero section, namely formula_30 which assigns to each point formula_31 the zero element formula_32 in the fiber over formula_31.
Blowing up submanifolds in complex manifolds.
More generally, one can blow up any codimension-"k" complex submanifold "Z" of C"n". Suppose that "Z" is the locus of the equations formula_34, and let formula_35 be homogeneous coordinates on P"k" - 1. Then the blow-up formula_36 is the locus of the equations formula_37 for all "i" and "j", in the space C"n" × P"k" - 1.
More generally still, one can blow up any submanifold of any complex manifold "X" by applying this construction locally. The effect is, as before, to replace the blow-up locus "Z" with the exceptional divisor "E". In other words, the blow-up map
is a birational mapping which, away from "E", induces an isomorphism, and, on "E", a locally trivial fibration with fiber P"k" - 1. Indeed, the restriction formula_39 is naturally seen as the projectivization of the normal bundle of "Z" in "X".
Since "E" is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that "E" intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; "E" is the only smooth complex representative of its homology class in formula_40. (Suppose "E" could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of "E".) This is why the divisor is called exceptional.
Let "V" be some submanifold of "X" other than "Z". If "V" is disjoint from "Z", then it is essentially unaffected by blowing up along "Z". However, if it intersects "Z", then there are two distinct analogues of "V" in the blow-up formula_40. One is the proper (or strict) transform, which is the closure of formula_42; its normal bundle in formula_40 is typically different from that of "V" in "X". The other is the total transform, which incorporates some or all of "E"; it is essentially the pullback of "V" in cohomology.
Blowing up schemes.
To pursue blow-up in its greatest generality, let "X" be a scheme, and let formula_44 be a coherent sheaf of ideals on "X". The blow-up of "X" with respect to formula_44 is a scheme formula_46 along with a morphism
such that formula_48 is an invertible sheaf, characterized by this universal property: for any morphism "f": "Y" → "X" such that formula_49 is an invertible sheaf, "f" factors uniquely through π.
Notice that
has this property; this is how the blow-up is constructed. Here "Proj" is the Proj construction on graded sheaves of commutative rings.
Exceptional divisors.
The exceptional divisor of a blowup formula_51 is the subscheme defined by the inverse image of the ideal sheaf formula_44, which is sometimes denoted formula_53. It follows from the definition of the blow up in terms of Proj that this subscheme "E" is defined by the ideal sheaf formula_54. This ideal sheaf is also the relative formula_55 for π.
π is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of π. That is, π may be an isomorphism on "E". This happens, for example, in the trivial situation where formula_44 is already an invertible sheaf. In particular, in such cases the morphism π does not determine the exceptional divisor. Another situation where the exceptional locus can be strictly smaller than the exceptional divisor is when "X" has singularities. For instance, let "X" be the affine cone over . "X" can be given as the vanishing locus of in A4. The ideals and define two planes, each of which passes through the vertex of "X". Away from the vertex, these planes are hypersurfaces in "X", so the blowup is an isomorphism there. The exceptional locus of the blowup of either of these planes is therefore centered over the vertex of the cone, and consequently it is strictly smaller than the exceptional divisor.
Further examples.
Blowups of linear subspaces.
Let formula_57 be -dimensional projective space. Fix a linear subspace of codimension . There are several explicit ways to describe the blowup of formula_57 along . Suppose that formula_57 has coordinates formula_60. After changing coordinates, we may assume that formula_61. The blowup may be embedded in formula_62. Let formula_63 be coordinates on the second factor. Because is defined by a regular sequence, the blowup is determined by the vanishing of the two-by-two minors of the matrix
formula_64
This system of equations is equivalent to asserting that the two rows are linearly dependent. A point formula_65 is in if and only if, when its coordinates are substituted in the first row of the matrix above, that row is zero. In this case, there are no conditions on . If, however, that row is non-zero, then linear dependence implies that the second row is a scalar multiple of the first and therefore that there is a unique point formula_66 such that formula_67 is in the blowup.
This blowup can also be given a synthetic description as the incidence correspondence
formula_68
where formula_69 denotes the Grassmannian of formula_70-dimensional subspaces in formula_57. To see the relation with the previous coordinatization, observe that the set of all formula_72 that contain is isomorphic to a projective space formula_73. This is because each subspace is the linear join of and a point not in , and two points and determine the same if and only if they have the same image under the projection of formula_57 away from . Therefore the Grassmannian may be replaced by a copy of formula_73. When formula_76, there is only one subspace containing , the linear join of and . In the coordinates above, this is the case where formula_77 is not the zero vector. The case formula_78 corresponds to formula_77 being the zero vector, and in this case, any is allowed, that is, any containing is possible.
Blowing up intersections of curves scheme-theoretically.
Let formula_80 be generic homogeneous polynomials of degree formula_81 (meaning their associated projective varieties intersects at formula_82 points by Bézout's theorem). The following projective morphism of schemes gives a model of blowing up formula_83 at formula_82 points:formula_85
Looking at the fibers explains why this is true: if we take a point formula_86 then the pullback diagram
formula_87
tells us the fiber is a point whenever formula_88 or formula_89 and the fiber is formula_90 if formula_91.
Related constructions.
In the blow-up of C"n" described above, there was nothing essential about the use of complex numbers; blow-ups can be performed over any field. For example, the "real" blow-up of R2 at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere S2 results in the real projective plane.
Deformation to the normal cone is a blow-up technique used to prove many results in algebraic geometry. Given a scheme "X" and a closed subscheme "V", one blows up
Then
is a fibration. The general fiber is naturally isomorphic to "X", while the central fiber is a union of two schemes: one is the blow-up of "X" along "V", and the other is the normal cone of "V" with its fibers completed to projective spaces.
Blow-ups can also be performed in the symplectic category, by endowing the symplectic manifold with a compatible almost complex structure and proceeding with a complex blow-up. This makes sense on a purely topological level; however, endowing the blow-up with a symplectic form requires some care, because one cannot arbitrarily extend the symplectic form across the exceptional divisor "E". One must alter the symplectic form in a neighborhood of "E", or perform the blow-up by cutting out a neighborhood of "Z" and collapsing the boundary in a well-defined way. This is best understood using the formalism of symplectic cutting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of symplectic summation, is the symplectic analogue of deformation to the normal cone along a smooth divisor.


Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.
A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and
column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.
Specifically, a matrix is in row echelon form if
Some texts add the condition that the leading coefficient must be 1.
These two conditions imply that all entries in a column below a leading coefficient are zeros.
The following is an example of a 3×5 matrix in row echelon form, which is not in "reduced" row echelon form (see below):
Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.
Reduced row echelon form.
A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it. For a given matrix, despite the row echelon form not being unique, all row echelon forms and the reduced row echelon form have the same number of zero rows and the pivots are located in the same indices.
This is an example of a matrix in reduced row echelon form, which shows that the left part of the matrix is not always an identity matrix:
For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be calculated using Euclidean division and without introducing any rational number or denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer coefficients.
Transformation to row echelon form.
By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to row echelon form. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix.
The resulting echelon form is not unique; any matrix that is in echelon form can be put in an (equivalent) echelon form by adding a scalar multiple of a row to one of the above rows, for example:
However, every matrix has a unique "reduced" row echelon form. In the above example, the reduced row echelon form can be found as
This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix.
Systems of linear equations.
A system of linear equations is said to be in "row echelon form" if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in "reduced row echelon form" or in "canonical form" if its augmented matrix is in reduced row echelon form.
The canonical form may be viewed as an explicit solution of the linear system. In fact, the system is inconsistent if and only if one of the equations of the canonical form is reduced to 0 = 1. Otherwise, regrouping in the right hand side all the terms of the equations but the leading ones, expresses the variables corresponding to the pivots as constants or linear functions of the other variables, if any.
Pseudocode for reduced row echelon form.
The following pseudocode converts a matrix into a reduced row echelon form:
 function ToReducedRowEchelonForm(Matrix M) is
 "lead" := 0
 "rowCount" := the number of rows in M
 "columnCount" := the number of columns in M
 for 0 ≤ "r" < "rowCount" do
 if "columnCount" ≤ "lead" then
 stop function
 end if
 "i" = "r"
 while M["i", "lead"] = 0 do
 "i" = "i" + 1
 if "rowCount" = "i" then
 "i" = "r"
 "lead" = "lead" + 1
 if "columnCount" = "lead" then
 stop function
 end if
 end if
 end while
 if "i" ≠ "r" then Swap rows "i" and "r"
 Divide row "r" by M["r", "lead"]
 for 0 ≤ "i" < "rowCount" do
 if "i" ≠ "r" do
 Subtract M[i, lead] multiplied by row "r" from row "i"
 end if
 end for
 "lead" = "lead" + 1
 end for
 end function
The following pseudocode converts the matrix to a row echelon form (not abbreviated):
 function ToRowEchelonForm(Matrix M) is
 "nr" := number of rows in M
 "nc" := number of columns in M
 for 0 ≤ r < nr do
 "allZeros" := true
 for 0 ≤ "c" < "nc" do
 if M["r", "c"] != 0 then
 "allZeros" := false
 exit for
 end if
 end for
 if "allZeros" = true then
 In M, swap row "r" with row "nr"
 "nr" := "nr" - 1
 end if
 end for
 "p" := 0
 while "p" < "nr" and "p" < "nc" do
 label nextPivot:
 "r" := 1
 while M["p", "p"] = 0 do
 if ("p" + "r") <= "nr" then
 "p" := "p" + 1
 goto nextPivot
 end if
 In M, swap row "p" with row ("p" + "r")
 "r" := "r" + 1
 end while
 for 1 ≤ "r" < ("nr" - "p") do
 if M["p" + "r", "p"] != 0 then
 "x" := -M["p" + "r", "p"] / M["p", "p"]
 for "p" ≤ "c" < "nc" do
 M["p" + "r", "c"] := M["p" , "c"] * "x" + M["p" + "r", "c"]
 end for
 end if
 end for
 "p" := "p" + 1
 end while
 end function


Algebraic cycle

In mathematics, an algebraic cycle on an algebraic variety "V" is a formal linear combination of subvarieties of "V". These are the part of the algebraic topology of "V" that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.
The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space.
While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For example, every curve has a constant "N" such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most "N". David Mumford proved that, on a smooth complete complex algebraic surface "S" with positive geometric genus, the analogous statement for the group formula_1 of rational equivalence classes of codimension two cycles in "S" is false. The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group formula_2 contains transcendental information, and in effect Mumford's theorem implies that, despite formula_1 having a purely algebraic definition, it shares transcendental information with formula_2. Mumford's theorem has since been greatly generalized.
The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millenium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles yield enough cycles to construct his category of motives and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely, Alexander Beilinson proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to algebraic "K"-theory by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of "K"-theory sheaves.
Definition.
Let "X" be a scheme which is finite type over a field "k". An algebraic "r"-cycle on "X" is a formal linear combination
of "r"-dimensional closed integral "k"-subschemes of "X". The coefficient "n""i" is the "multiplicity" of "V""i". The set of all "r"-cycles is the free abelian group
where the sum is over closed integral subschemes "V" of "X". The groups of cycles for varying "r" together form a group
This is called the group of algebraic cycles, and any element is called an algebraic cycle. A cycle is effective or positive if all its coefficients are non-negative.
Closed integral subschemes of "X" are in one-to-one correspondence with the scheme-theoretic points of "X" under the map that, in one direction, takes each subscheme to its generic point, and in the other direction, takes each point to the unique reduced subscheme supported on the closure of the point. Consequently formula_8 can also be described as the free abelian group on the points of "X".
A cycle formula_9 is rationally equivalent to zero, written formula_10, if there are a finite number of formula_11-dimensional subvarieties formula_12 of formula_13 and non-zero rational functions formula_14 such that formula_15, where formula_16 denotes the divisor of a rational function on "W""i". The cycles rationally equivalent to zero are a subgroup formula_17, and the group of "r"-cycles modulo rational equivalence is the quotient
This group is also denoted formula_19. Elements of the group
are called cycle classes on "X". Cycle classes are said to be effective or positive if they can be represented by an effective cycle.
If "X" is smooth, projective, and of pure dimension "N", the above groups are sometimes reindexed cohomologically as
and
In this case, formula_23 is called the Chow ring of "X" because it has a multiplication operation given by the intersection product.
There are several variants of the above definition. We may substitute another ring for integers as our coefficient ring. The case of rational coefficients is widely used. Working with families of cycles over a base, or using cycles in arithmetic situations, requires a relative setup. Let formula_24, where "S" is a regular Noetherian scheme. An "r"-cycle is a formal sum of closed integral subschemes of "X" whose relative dimension is "r"; here the relative dimension of formula_25 is the transcendence degree of formula_26 over formula_27 minus the codimension of formula_28 in "S".
Rational equivalence can also be replaced by several other coarser equivalence relations on algebraic cycles. Other equivalence relations of interest include "algebraic equivalence", "homological equivalence" for a fixed cohomology theory (such as singular cohomology or étale cohomology), "numerical equivalence", as well as all of the above modulo torsion. These equivalence relations have (partially conjectural) applications to the theory of motives.
Flat pullback and proper pushforward.
There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let "f" : "X" → "X'" be a map of varieties.
If "f" is flat of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety "Y'" ⊂ "X'":
which by assumption has the same codimension as "Y′".
Conversely, if "f" is proper, for "Y" a subvariety of "X" the pushforward is defined to be
where "n" is the degree of the extension of function fields ["k"("Y") : "k"("f"("Y"))] if the restriction of "f" to "Y" is finite and 0 otherwise.
By linearity, these definitions extend to homomorphisms of abelian groups
(the latter by virtue of the convention) are homomorphisms of abelian groups. See Chow ring for a discussion of the functoriality related to the ring structure.


Decomposition theorem

In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.
Statement.
Decomposition for smooth proper maps.
The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map formula_1 of relative dimension "d" between two projective varieties
Here formula_3 is the fundamental class of a hyperplane section, formula_4 is the direct image (pushforward) and formula_5 is the "n"-th derived functor of the direct image. This derived functor measures the "n"-th cohomologies of formula_6, for formula_7.
In fact, the particular case when "Y" is a point, amounts to the isomorphism
This hard Lefschetz isomorphism induces canonical isomorphisms
Moreover, the sheaves formula_10 appearing in this decomposition are local systems, i.e., locally free sheaves of Q-vector spaces, which are moreover semisimple, i.e., a direct sum of local systems without nontrivial local subsystems.
Decomposition for proper maps.
The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map formula_1 between varieties. In a nutshell, the results above remain true when the notion of local systems is replaced by perverse sheaves.
The hard Lefschetz theorem above takes the following form: there is an isomorphism in the derived category of sheaves on "Y":
where formula_13 is the total derived functor of formula_4 and formula_15 is the "i"-th truncation with respect to the "perverse" t-structure.
Moreover, there is an isomorphism
where the summands are semi-simple perverse-sheaves, meaning they are direct sums of push-forwards of intersection cohomology sheaves.
If "X" is not smooth, then the above results remain true when formula_17 is replaced by the intersection cohomology complex formula_18.
Proofs.
The decomposition theorem was first proved by Beilinson, Bernstein, and Deligne. Their proof is based on the usage of weights on l-adic sheaves in positive characteristic. A different proof using mixed Hodge modules was given by Saito. A more geometric proof, based on the notion of semismall maps was given by de Cataldo and Migliorini.
For semismall maps, the decomposition theorem also applies to Chow motives.
Applications of the Decomposition Theorem.
Cohomology of a Rational Lefschetz Pencil.
Consider a rational morphism formula_19 from a smooth quasi-projective variety given by formula_20. If we set the vanishing locus of formula_21 as formula_22 then there is an induced morphism formula_23. We can compute the cohomology of formula_24 from the intersection cohomology of formula_25 and subtracting off the cohomology from the blowup along formula_22. This can be done using the perverse spectral sequence


Ragsdale conjecture

The Ragsdale conjecture is a mathematical conjecture that concerns the possible arrangements of real algebraic curves embedded in the projective plane. It was proposed by Virginia Ragsdale in her dissertation in 1906 and was disproved in 1979. It has been called "the oldest and most famous conjecture on the topology of real algebraic curves".
Formulation of the conjecture.
Ragsdale's dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. The dissertation was a treatment of Hilbert's sixteenth problem, which had been proposed by Hilbert in 1900, along with 22 other unsolved problems of the 19th century; it is one of the handful of Hilbert's problems that remains wholly unresolved. Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type, along with the basis of evidence.
Conjecture.
Ragsdale's main conjecture is as follows.
Assume that an algebraic curve of degree 2"k" contains "p" even and "n" odd ovals. Ragsdale conjectured that
She also posed the inequality
and showed that the inequality could not be further improved. This inequality was later proved by Petrovsky.
Disproving the conjecture.
The conjecture was held of very high importance in the field of real algebraic geometry for most of the twentieth century. Later, in 1980, Oleg Viro introduced a technique known as "patchworking algebraic curves" and used to generate a counterexample to the conjecture.
In 1993, Ilia Itenberg produced additional counterexamples to the Ragsdale conjecture, so Viro and Itenberg wrote a paper in 1996 discussing their work on disproving the conjecture using the "patchworking" technique.
The problem of finding a sharp upper bound remains unsolved.


Spark (mathematics)

Definition.
In mathematics, specifically in linear algebra, the spark of a formula_1 matrix formula_2 is the smallest number formula_3 such that there exists a set of formula_3 columns in formula_2 which are linearly dependent. Formally,
where formula_6 is a nonzero vector and formula_7 denotes its number of nonzero coefficients.
If all the columns are linearly independent, formula_8 is usually defined to be formula_9.
By contrast, the rank of a matrix is the largest number formula_3 such that some set of formula_3 columns of formula_2 is linearly independent.
Example.
Consider the following matrix formula_2.
formula_14
The spark of this matrix equals 3 because:
The first three columns are linearly dependent because formula_18.
Properties.
If formula_19, the following simple properties hold for the spark of a formula_1 matrix formula_2:
Criterion for uniqueness of sparse solutions.
The spark yields a simple criterion for uniqueness of sparse solutions of linear equation systems.
Given a linear equation system formula_26. If this system has a solution formula_27 that satisfies formula_28, then this solution is the sparsest possible solution. Here formula_29 denotes the number of nonzero entries of the vector formula_27.
Lower bound in terms of dictionary coherence.
If the columns of the matrix formula_2 are normalized to unit norm, we can lower bound the spark of the matrix in terms of the dictionary coherence:
The dictionary coherence formula_33 is defined as the maximum correlation between any two columns, i.e.
Applications.
The minimum distance of a linear code equals the spark of its parity-check matrix.
The concept of the spark is also of use in the theory of compressive sensing, where requirements on the spark of the measurement matrix are used to ensure stability and consistency of various estimation techniques. It is also known in matroid theory as the girth of the vector matroid associated with the columns of the matrix. The spark of a matrix is NP-hard to compute.


Little q-Laguerre polynomials

In mathematics, the little "q"-Laguerre polynomials "p""n"("x";"a"|"q") or Wall polynomials "W""n"("x"; "b","q") are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by . (The term "Wall polynomial" is also used for an unrelated Wall polynomial in the theory of classical groups.)
 give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by


Joel Lee Brenner

Joel Lee Brenner ( – ) was an American mathematician who specialized in matrix theory, linear algebra, and group theory. He is known as the translator of several popular Russian texts. He was a teaching professor at some dozen colleges and universities and was a Senior Mathematician at Stanford Research Institute from 1956 to 1968. He published over one hundred scholarly papers, 35 with coauthors, and wrote book reviews.
Academic career.
In 1930 Brenner earned a B.A. degree with major in chemistry from Harvard University. In graduate study there he was influenced by Hans Brinkmann, Garrett Birkhoff, and Marshall Stone. He was granted the Ph.D. in February 1936. Brenner later described some of his reminiscences of his student days at Harvard and of the state of American mathematics in the 1930s in an article for American Mathematical Monthly.
In 1951 Brenner published his findings about matrices with quaternion entries. He developed the idea of a characteristic root of a quaternion matrix (an eigenvalue) and shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix.
In 1956 he became a Senior Mathematician at Stanford Research Institute.
Brenner, in collaboration with Donald W. Bushaw and S. Evanusa, assisted in the translation and revision of Felix Gantmacher's "Applications of the Theory of Matrices" (1959).
Brenner translated Nikolaj Nikolaevič Krasovskii's book "Stability of motion: applications of Lyapunov's second method to differential systems and equations with delay" (1963). He also translated and edited the book "Problems in differential equations" by Aleksei Fedorovich Filippov.
Brenner translated "Problems in Higher Algebra" by D. K. Faddeev and I.S. Sominiski. The exercises in this book covered complex numbers, roots of unity, as well as some linear algebra and abstract algebra.
In 1959 Brenner generalized propositions by Alexander Ostrowski and G. B. Price on minors of a diagonally dominant matrix. His work is credited with stimulating a reawakening of interest in the permanent of a matrix.
One of the challenges in linear algebra is to find the eigenvalues and eigenvectors of a square matrix of complex numbers. In 1931 S. A. Gershgorin described geometric bounds on the eigenvectors in terms of the matrix elements. This result known as the Gershgorin circle theorem has been used as a basis for extension. In 1964 Brenner reported on "Theorems of Gersgorin Type". In 1967 at University of Wisconsin—Madison, working in the Mathematics Research Center, he produced a technical report "New root-location theorems for partitioned matrices".
In 1968 Brenner, following Alston Householder, published "Gersgorin theorems by Householder’s proof". In 1970 he published the survey article (21 references) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices". The article was extended with "Some determinantal identities".
In 1971 Brenner extended his geometry of the spectrum of a square complex matrix deeper into abstract algebra with his paper "Regularity theorems and Gersgorin theorems for matrices over rings with valuation". He writes, "Theorems can be extended to non-commutative domains, in particular to quaternion matrices. Secondly, the ring of polynomials has a valuation ... a different type of regularity ..."
Collaborations.
Joel Lee Brenner was a member of the American Mathematical Society from 1936.
Beasley relates that he
In 1981 Brenner and Roger Lyndon collaborated to polish an idea due to H. W. Kuhn for proving the fundamental theorem of algebra. In the solution by Eric S. Rosenthal to a problem in the "American Mathematical Monthly" posted by Harry D. Ruderman, Kuhn's work from 1974 was cited. A query was made and prompted an article by Brenner and Lyndon. The version of the fundamental theorem stated was as follows:
Brenner ultimately acquired 35 coauthors in his publications.
Alternating group.
Given an ordered set Ω with "n" elements, the even permutations on it determine the alternating group An. In 1960 Brenner proposed the following research problem in group theory: For which An does there exist an element "a"n such that every element "g" is similar to a commutator of "a"n? Brenner states that the property is true for 4 < "n" < 10; in symbols it may be expressed
The alternating groups are simple groups, and in 1971 Brenner began a series of articles titled "Covering theorems for finite simple groups". He was interested in the cycle type of cyclic permutations, and when An ⊂ "C C", where "C" is a conjugacy class of a certain type.
In 1977 he posed the question, "What permutations in An can be expressed as a product of permutations of periods k and l" ?
Works.
In 1987 Linear Algebra and its Applications published a list of 111 articles by J.L. Brenner, and the four books he translated.


Johnson graph

Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph formula_1 are the formula_2-element subsets of an formula_3-element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains formula_4-elements. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson.
Automorphism group.
There is a distance-transitive subgroup of formula_24 isomorphic to formula_25. In fact, formula_26, unless formula_27; otherwise, formula_28.
Intersection array.
As a consequence of being distance-transitive, formula_1 is also distance-regular. Letting formula_30 denote its diameter, the intersection array of formula_1 is given by
where:
It turns out that unless formula_1 is formula_35, its intersection array is not shared with any other distinct distance-regular graph; the intersection array of formula_35 is shared with three other distance-regular graphs that are not Johnson graphs.
Johnson scheme.
The Johnson graph formula_1 is closely related to the Johnson scheme, an association scheme in which each pair of -element sets is associated with a number, half the size of the symmetric difference of the two sets. The Johnson graph has an edge for every pair of sets at distance one in the association scheme, and the distances in the association scheme are exactly the shortest path distances in the Johnson graph.
The Johnson scheme is also related to another family of distance-transitive graphs, the odd graphs, whose vertices are formula_2-element subsets of an formula_43-element set and whose edges correspond to disjoint pairs of subsets.
Open Problems.
The vertex-expansion properties of Johnson graphs, as well as the structure of the corresponding extremal sets of vertices of a given size, are not fully understood. However, an asymptotically tight lower-bound on expansion of large sets of vertices was recently obtained.
In general, determining the chromatic number of a Johnson graph is an open problem.


Simple (abstract algebra)

In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:
The general pattern is that the structure admits no non-trivial congruence relations.
The term is used differently in semigroup theory. A semigroup is said to be "simple" if it has no nontrivial
ideals, or equivalently, if Green's relation "J" is
the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may
have nontrivial congruences. A semigroup with no nontrivial congruences is called "congruence simple".


Reinhardt Kiehl

Reinhardt Kiehl (born 31 May 1935 in Herne, North Rhine-Westphalia) is a German mathematician.
From 1955, Kiehl studied mathematics, physics and astronomy at the University of Göttingen and the University of Heidelberg. He received in 1965 his Ph.D. (promotion) under Friedrich Karl Schmidt at Heidelberg University with thesis "Äquivalenzrelationen in analytischen Räumen". He was from 1966 to 1968 a research assistant and in 1968–1969 a docent at the University of Münster, where he received in 1968 his habilitation. From 1969 to 1972 he was a professor ordinarius at the Goethe-Universität Frankfurt am Main. From 1972 he was a professor ordinarius at the University of Mannheim, where he retired in 2003 as professor emeritus.
His research deals with algebraic and arithmetic geometry and non-archimedean function theory. He wrote with Eberhard Freitag a textbook on the Weil conjectures and étale cohomology. In 1970 Kiehl was an Invited Speaker at the ICM in Nice with talk "Grauertsche Kohärenzsätze für stetige und differenzierbare Familien komplexer Räume".


Homogeneous coordinate ring

In algebraic geometry, the homogeneous coordinate ring "R" of an algebraic variety "V" given as a subvariety of projective space of a given dimension "N" is by definition the quotient ring
where "I" is the homogeneous ideal defining "V", "K" is the algebraically closed field over which "V" is defined, and
is the polynomial ring in "N" + 1 variables "X""i". The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.
Formulation.
Since "V" is assumed to be a variety, and so an irreducible algebraic set, the ideal "I" can be chosen to be a prime ideal, and so "R" is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction.
The "irrelevant ideal" "J" generated by all the "X""i" corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space.
The projective Nullstellensatz gives a bijective correspondence between projective varieties and homogeneous ideals "I" not containing "J".
Resolutions and syzygies.
In application of homological algebra techniques to algebraic geometry, it has been traditional since David Hilbert (though modern terminology is different) to apply free resolutions of "R", considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal "I". In a classical perspective, such generators are simply the equations one writes down to define "V". If "V" is a hypersurface there need only be one equation, and for complete intersections the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of canonical curves and the equations defining abelian varieties, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of elimination theory in its classical form, in which reduction modulo "I" is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
There are for general reasons free resolutions of "R" as graded module over "K"["X"0, "X"1, "X"2, ..., "X""N"]. A resolution is defined as "minimal" if the image in each module morphism of free modules
in the resolution lies in "JF""i" − 1, where "J" is the irrelevant ideal. As a consequence of Nakayama's lemma, φ then takes a given basis in "F""i" to a minimal set of generators in "F""i" − 1. The concept of "minimal free resolution" is well-defined in a strong sense: unique up to isomorphism of chain complexes and occurring as a direct summand in any free resolution. Since this complex is intrinsic to "R", one may define the graded Betti numbers β"i, j" as the number of grade-"j" images coming from "F""i" (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). In other words, weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution. The properties of these invariants of "V" in a given projective embedding poses active research questions, even in the case of curves.
There are examples where the minimal free resolution is known explicitly. For a rational normal curve it is an Eagon–Northcott complex. For elliptic curves in projective space the resolution may be constructed as a mapping cone of Eagon–Northcott complexes.
Regularity.
The Castelnuovo–Mumford regularity may be read off the minimum resolution of the ideal "I" defining the projective variety. In terms of the imputed "shifts" "a""i", "j" in the "i"-th module "F""i", it is the maximum over "i" of the "a""i", "j" − "i"; it is therefore small when the shifts increase only by increments of 1 as we move to the left in the resolution (linear syzygies only).
Projective normality.
The variety "V" in its projective embedding is projectively normal if "R" is integrally closed. This condition implies that "V" is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions. Another equivalent condition is in terms of the linear system of divisors on "V" cut out by the dual of the tautological line bundle on projective space, and its "d"-th powers for "d" = 1, 2, 3, ... ; when "V" is non-singular, it is projectively normal if and only if each such linear system is a complete linear system. Alternatively one can think of the dual of the tautological line bundle as the Serre twist sheaf "O"(1) on projective space, and use it to twist the structure sheaf "O""V" any number of times, say "k" times, obtaining a sheaf "O""V"("k"). Then "V" is called "k"-normal if the global sections of "O"("k") map surjectively to those of "O""V"("k"), for a given "k", and if "V" is 1-normal it is called linearly normal. A non-singular variety is projectively normal if and only if it is "k"-normal for all "k" ≥ 1. Linear normality may also be expressed geometrically: "V" as projective variety cannot be obtained by an isomorphic linear projection from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough Veronese mappings to reduce it to conditions of linear normality.
Looking at the issue from the point of view of a given very ample line bundle giving rise to the projective embedding of "V", such a line bundle (invertible sheaf) is said to be normally generated if "V" as embedded is projectively normal. Projective normality is the first condition "N"0 of a sequence of conditions defined by Green and Lazarsfeld. For this
is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken. Condition "N"p applied to the first "p" graded Betti numbers, requiring they vanish when "j" > "i" + 1. For curves Green showed that condition "N""p" is satisfied when deg("L") ≥ 2"g" + 1 + "p", which for "p" = 0 was a classical result of Guido Castelnuovo.


Classical involution theorem

In mathematical finite group theory, the classical involution theorem of classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. extended the classical involution theorem to groups of finite Morley rank.
A classical involution "t" of a finite group "G" is an involution whose centralizer has a subnormal subgroup containing "t" with quaternion Sylow 2-subgroups.


Graph of groups

In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups.
There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabiliser subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.
Definition.
A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends.
Fundamental group.
Let be a spanning tree for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of with the following relations:
This definition is independent of the choice of .
The benefit in defining the fundamental groupoid of a graph of groups, as shown by , is that it is defined independently of base point or tree. Also there is proved there a nice normal form for the elements of the fundamental groupoid. This includes normal form theorems for a free product with amalgamation and for an HNN extension .
Structure theorem.
Let be the fundamental group corresponding to the spanning tree . For every vertex and edge , and can be identified with their images in . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces and respectively. This graph is a tree, called the universal covering tree, on which acts. It admits the graph as fundamental domain. The graph of groups given by the stabiliser subgroups on the fundamental domain corresponds to the original graph of groups.
Generalisations.
The simplest possible generalisation of a graph of groups is a 2-dimensional complex of groups. These are modeled on orbifolds arising from cocompact properly discontinuous actions of discrete groups on 2-dimensional simplicial complexes that have the structure of CAT(0) spaces. The quotient of the simplicial complex has finite stabiliser groups attached to vertices, edges and triangles together with monomorphisms for every inclusion of simplices. A complex of groups is said to be developable if it arises as the quotient of a CAT(0) simplicial complex. Developability is a non-positive curvature condition on the complex of groups: it can be verified locally by checking that all circuits occurring in the links of vertices have length at least six. Such complexes of groups originally arose in the theory of 2-dimensional Bruhat–Tits buildings; their
general definition and continued study have been inspired by the ideas of Gromov.


Subrepresentation

In representation theory, a subrepresentation of a representation formula_1 of a group "G" is a representation formula_2 such that "W" is a vector subspace of "V" and formula_3.
A finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations.
If formula_1 is a representation of "G", then there is the trivial subrepresentation:


Word (group theory)

In group theory, a word is any written product of group elements and their inverses. For example, if "x", "y" and "z" are elements of a group "G", then "xy", "z"−1"xzz" and "y"−1"zxx"−1"yz"−1 are words in the set {"x", "y", "z"}. Two different words may evaluate to the same value in "G", or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory.
Definition.
Let "G" be a group, and let "S" be a subset of "G". A word in "S" is any expression of the form
where "s"1...,"sn" are elements of "S" and each "εi" is ±1. The number "n" is known as the length of the word.
Each word in "S" represents an element of "G", namely the product of the expression. By convention, the identity (unique) element can be represented by the empty word, which is the unique word of length zero.
Notation.
When writing words, it is common to use exponential notation as an abbreviation. For example, the word
could be written as
This latter expression is not a word itself—it is simply a shorter notation for the original.
When dealing with long words, it can be helpful to use an overline to denote inverses of elements of "S". Using overline notation, the above word would be written as follows:
Words and presentations.
A subset "S" of a group "G" is called a generating set if every element of "G" can be represented by a word in "S". If "S" is a generating set, a relation is a pair of words in "S" that represent the same element of "G". These are usually written as equations, e.g. formula_5
A set formula_6 of relations defines "G" if every relation in "G" follows logically from those in formula_6, using the axioms for a group. A presentation for "G" is a pair formula_8, where "S" is a generating set for "G" and formula_6 is a defining set of relations.
For example, the Klein four-group can be defined by the presentation
Here 1 denotes the empty word, which represents the identity element.
When "S" is not a generating set for "G", the set of elements represented by words in "S" is a subgroup of "G". This is known as the subgroup of "G" generated by "S", and is usually denoted formula_11. It is the smallest subgroup of "G" that contains the elements of "S".
Reduced words.
Any word in which a generator appears next to its own inverse ("xx"−1 or "x"−1"x") can be simplified by omitting the redundant pair:
This operation is known as reduction, and it does not change the group element represented by the word. (Reductions can be thought of as relations that follow from the group axioms.)
A reduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions:
The result does not depend on the order in which the reductions are performed.
If "S" is any set, the free group over "S" is the group with presentation formula_14. That is, the free group over "S" is the group generated by the elements of "S", with no extra relations. Every element of the free group can be written uniquely as a reduced word in "S".
A word is cyclically reduced if and only if every cyclic permutation of the word is reduced.
Normal forms.
A normal form for a group "G" with generating set "S" is a choice of one reduced word in "S" for each element of "G". For example:
Operations on words.
The product of two words is obtained by concatenation:
Even if the two words are reduced, the product may not be.
The inverse of a word is obtained by inverting each generator, and switching the order of the elements:
The product of a word with its inverse can be reduced to the empty word:
You can move a generator from the beginning to the end of a word by conjugation:
The word problem.
Given a presentation formula_19 for a group "G", the word problem is the algorithmic problem of deciding, given as input two words in "S", whether they represent the same element of "G". The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group "G" such that the word problem for "G" is undecidable.


Racah polynomials

In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.
The Racah polynomials were first defined by and are given by
 introduced the "q"-Racah polynomials defined in terms of basic hypergeometric functions by
They are sometimes given with changes of variables as


Dickson polynomial

In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials.
Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials.
Dickson polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of "permutation polynomials"; polynomials acting as permutations of finite fields.
Definition.
First kind.
For integer and in a commutative ring with identity (often chosen to be the finite field ) the Dickson polynomials (of the first kind) over are given by
The first few Dickson polynomials are
They may also be generated by the recurrence relation for ,
with the initial conditions and .
Second kind.
The Dickson polynomials of the second kind, , are defined by
They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind.
The first few Dickson polynomials of the second kind are
They may also be generated by the recurrence relation for ,
with the initial conditions and .
Properties.
The are the unique monic polynomials satisfying the functional equation
where and .
They also satisfy a composition rule,
The also satisfy a functional equation
for , , with and .
The Dickson polynomial is a solution of the ordinary differential equation
and the Dickson polynomial is a solution of the differential equation
Their ordinary generating functions are
Links to other polynomials.
By the recurrence relation above, Dickson polynomials are Lucas sequences. Specifically, for , the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of the second kind are Lucas polynomials.
By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative.
formula_13
formula_14
Permutation polynomials and Dickson polynomials.
A permutation polynomial (for a given finite field) is one that acts as a permutation of the elements of the finite field.
The Dickson polynomial (considered as a function of with α fixed) is a permutation polynomial for the field with elements if and only if is coprime to .
 proved that any integral polynomial that is a permutation polynomial for infinitely many prime fields is a composition of Dickson polynomials and linear polynomials (with rational coefficients). This assertion has become known as Schur's conjecture, although in fact Schur did not make this conjecture. Since Fried's paper contained numerous errors, a corrected account was given by , and subsequently gave a simpler proof along the lines of an argument due to Schur.
Further, proved that any permutation polynomial over the finite field whose degree is simultaneously coprime to and less than must be a composition of Dickson polynomials and linear polynomials.
Generalization.
Dickson polynomials of both kinds over finite fields can be thought of as initial members of a sequence of generalized Dickson polynomials referred to as Dickson polynomials of the th kind. Specifically, for with for some prime and any integers and , the th Dickson polynomial of the th kind over , denoted by , is defined by
and
 and , showing that this definition unifies and generalizes the original polynomials of Dickson.
The significant properties of the Dickson polynomials also generalize:


Balanced matrix

In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd order having all row sums and all column sums equal to 2.
Balanced matrices are studied in linear programming. The importance of balanced matrices comes from the fact that the solution to a linear programming problem is integral if its matrix of coefficients is balanced and its right hand side or its objective vector is an all-one vector. In particular, if one searches for an integral solution to a linear program of this kind, it is not necessary to explicitly solve an integer linear program, but it suffices to find an optimal vertex solution of the linear program itself.
As an example, the following matrix is a balanced matrix:
Characterization by forbidden submatrices.
Equivalent to the definition, a 0-1 matrix is balanced if and only if it does not contain a submatrix that is the incidence matrix of any "odd cycle" (a cycle graph of odd order).
Therefore, the only three by three 0-1 matrix that is not balanced is (up to permutation of the rows and columns) the following incidence matrix of a cycle graph of order 3:
The following matrix is the incidence matrix of a cycle graph of order 5:
The above characterization implies that any matrix containing formula_4 or formula_5 (or the incidence matrix of any other odd cycle) as a submatrix, is not balanced.
Connection to other matrix classes.
Every balanced matrix is a perfect matrix.
More restricting than the notion of balanced matrices is the notion of "totally balanced matrices". A 0-1 matrix is called totally balanced if it does not contain a square submatrix having no repeated columns and all row sums and all column sums equal to 2. Equivalently, a matrix is totally balanced if and only if it does not contain a submatrix that is the incidence matrix of any cycle (no matter if of odd or even order). This characterization immediately implies that any totally balanced matrix is balanced.
Moreover, any 0-1 matrix that is totally unimodular is also balanced. The following matrix is a balanced matrix as it does not contain any submatrix that is the incidence matrix of an odd cycle:
Since this matrix is not totally unimodular (its determinant is -2), 0-1 totally unimodular matrices are a proper subset of balanced matrices.
For example, balanced matrices arise as the coefficient matrix in special cases of the set partitioning problem.
An alternative method of identifying some balanced matrices is through the subsequence count, where the subsequence count "SC" of any row s of matrix "A" is
If a 0-1 matrix "A" has SC("s") ≤ 1 for all rows "s" = 1, ..., "m", then "A" has a unique subsequence, is totally unimodular and therefore also balanced. Note that this condition is sufficient but not necessary for "A" to be balanced. In other words, the 0-1 matrices with SC("s") ≤ 1 for all rows "s" = 1, ..., "m" are a proper subset of the set of balanced matrices.
As a more general notion, a matrix where every entry is either 0, 1 or -1 is called balanced if in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of 4.


PROGRAMDATA (environment variable)


Helmut Wielandt

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Helmut Wielandt (19 December 1910 – 14 February 2001) was a German mathematician who worked on permutation groups.
He was born in Niedereggenen, Lörrach, Germany.
He gave a plenary lecture "Entwicklungslinien in der Strukturtheorie der endlichen Gruppen" (Lines of Development in the Structure Theory of Finite Groups) at the International Congress of Mathematicians (ICM) in 1958 at Edinburgh and was an Invited Speaker with talk "Bedingungen für die Konjugiertheit von Untergruppen endlicher Gruppen" (Conditions for the Conjugacy of Finite Groups) at the ICM in 1962 in Stockholm.
Publications.
Among his work in Algebra is an elegant proof of the Sylow Theorems (replacing an older cumbersome proof involving double cosets) that is in the standard textbooks on Abstract Algebra, i.e. Group Theory.


B-admissible representation

In mathematics, the formalism of "B"-admissible representations provides constructions of full Tannakian subcategories of the category of representations of a group "G" on finite-dimensional vector spaces over a given field "E". In this theory, "B" is chosen to be a so-called ("E", "G")-regular ring, i.e. an "E"-algebra with an "E"-linear action of "G" satisfying certain conditions given below. This theory is most prominently used in "p"-adic Hodge theory to define important subcategories of "p"-adic Galois representations of the absolute Galois group of local and global fields.
("E", "G")-rings and the functor "D".
Let "G" be a group and "E" a field. Let Rep("G") denote a non-trivial strictly full subcategory of the Tannakian category of "E"-linear representations of "G" on finite-dimensional vector spaces over "E" stable under subobjects, quotient objects, direct sums, tensor products, and duals.
An ("E", "G")-ring is a commutative ring "B" that is an "E"-algebra with an "E"-linear action of "G". Let "F" = "BG" be the "G"-invariants of "B". The covariant functor "DB" : Rep("G") → Mod"F" defined by
is "E"-linear (Mod"F" denotes the category of "F"-modules). The inclusion of "DB"(V) in "B" ⊗"EV" induces a homomorphism
called the comparison morphism.
Regular ("E", "G")-rings and "B"-admissible representations.
An ("E", "G")-ring "B" is called regular if
The third condition implies "F" is a field. If "B" is a field, it is automatically regular.
When "B" is regular,
with equality if, and only if, α"B,V" is an isomorphism.
A representation "V" ∈ Rep("G") is called "B"-admissible if α"B,V" is an isomorphism. The full subcategory of "B"-admissible representations, denoted Rep"B"("G"), is Tannakian.
If "B" has extra structure, such as a filtration or an "E"-linear endomorphism, then "DB"("V") inherits this structure and the functor "DB" can be viewed as taking values in the corresponding category.
Potentially "B"-admissible representations.
A potentially "B"-admissible representation captures the idea of a representation that becomes "B"-admissible when restricted to some subgroup of "G".


Totally real number field

In number theory, a number field "K" is called totally real if for each embedding of "K" into the complex numbers the image lies inside the real numbers. Equivalent conditions are that "K" is generated over Q by one root of an integer polynomial "P", all of the roots of "P" being real; or that the tensor product algebra of "K" with the real field, over Q, is isomorphic to a tensor power of R.
For example, quadratic fields "K" of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial "P" irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will "not" be totally real, although it is a field of real numbers.
The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.


Major index

In mathematics (and particularly in combinatorics), the major index of a permutation is the sum of the positions of the descents of the permutation. In symbols, the major index of the permutation "w" is
For example, if "w" is given in one-line notation by "w" = 351624 (that is, "w" is the permutation of {1, 2, 3, 4, 5, 6} such that "w"(1) = 3, "w"(2) = 5, etc.) then "w" has descents at positions 2 (from 5 to 1) and 4 (from 6 to 2) and so maj("w") = 2 + 4 = 6.
This statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of inversions. That is, the number of permutations of length "n" with "k" inversions is the same as the number of permutations of length "n" with major index equal to "k". (These numbers are known as "Mahonian numbers", also in honor of MacMahon.) In fact, a stronger result is true: the number of permutations of length "n" with major index "k" and "i" inversions is the same as the number of permutations of length "n" with major index "i" and "k" inversions, that is, the two statistics are equidistributed. For example, the number of permutations of length 4 with given major index and number of inversions is given in the table below.


Regular representation

In mathematics, and in particular the theory of group representations, the regular representation of a group "G" is the linear representation afforded by the group action of "G" on itself by translation.
One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation.
Finite groups.
For a finite group "G", the left regular representation λ (over a field "K") is a linear representation on the "K"-vector space "V" freely generated by the elements of "G", i. e. they can be identified with a basis of "V". Given "g" ∈ "G", λ"g" is the linear map determined by its action on the basis by left translation by "g", i.e.
For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given "g" ∈ "G", ρ"g" is the linear map on "V" determined by its action on the basis by right translation by "g"−1, i.e.
Alternatively, these representations can be defined on the "K"-vector space "W" of all functions . It is in this form that the regular representation is generalized to topological groups such as Lie groups.
The specific definition in terms of "W" is as follows. Given a function and an element "g" ∈ "G",
and
Significance of the regular representation of a group.
Every group "G" acts on itself by translations. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {"e"} of "G". The regular representation of "G", for a given field "K", is the linear representation made by taking this permutation representation as a set of basis vectors of a vector space over "K". The significance is that while the permutation representation doesn't decompose - it is transitive - the regular representation in general breaks up into smaller representations. For example, if "G" is a finite group and "K" is the complex number field, the regular representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity its dimension. The number of these irreducibles is equal to the number of conjugacy classes of "G".
The above fact can be explained by character theory. Recall that the character of the regular representation χ"(g)" is the number of fixed points of "g" acting on the regular representation "V". It means the number of fixed points χ"(g)" is zero when "g" is not "id" and |"G"| otherwise. Let "V" has the decomposition ⊕"a""i""V""i" where "V""i"'s are irreducible representations of "G" and "a""i"'s are the corresponding multiplicities. By character theory, the multiplicity "a""i" can be computed as
formula_5
which means the multiplicity of each irreducible representation is its dimension.
The article on group rings articulates the regular representation for finite groups, as well as showing how the regular representation can be taken to be a module.
Module theory point of view.
To put the construction more abstractly, the group ring "K"["G"] is considered as a module over itself. (There is a choice here of left-action or right-action, but that is not of importance except for notation.) If "G" is finite and the characteristic of K doesn't divide |"G"|, this is a semisimple ring and we are looking at its left (right) ring ideals. This theory has been studied in great depth. It is known in particular that the direct sum decomposition of the regular representation contains a representative of every isomorphism class of irreducible linear representations of "G" over "K". You can say that the regular representation is "comprehensive" for representation theory, in this case. The modular case, when the characteristic of "K" does divide |"G"|, is harder mainly because with "K"["G"] not semisimple, a representation can fail to be irreducible without splitting as a direct sum.
Structure for finite cyclic groups.
For a cyclic group "C" generated by "g" of order "n", the matrix form of an element of "K"["C"] acting on "K"["C"] by multiplication takes a distinctive form known as a "circulant matrix", in which each row is a shift to the right of the one above (in cyclic order, i.e. with the right-most element appearing on the left), when referred to the natural basis
When the field "K" contains a primitive n-th root of unity, one can diagonalise the representation of "C" by writing down "n" linearly independent simultaneous eigenvectors for all the "n"×"n" circulants. In fact if ζ is any "n"-th root of unity, the element
is an eigenvector for the action of "g" by multiplication, with eigenvalue
and so also an eigenvector of all powers of "g", and their linear combinations.
This is the explicit form in this case of the abstract result that over an algebraically closed field "K" (such as the complex numbers) the regular representation of "G" is completely reducible, provided that the characteristic of "K" (if it is a prime number "p") doesn't divide the order of "G". That is called "Maschke's theorem". In this case the condition on the characteristic is implied by the existence of a "primitive" "n"-th root of unity, which cannot happen in the case of prime characteristic "p" dividing "n".
Circulant determinants were first encountered in nineteenth century mathematics, and the consequence of their diagonalisation drawn. Namely, the determinant of a circulant is the product of the "n" eigenvalues for the "n" eigenvectors described above. The basic work of Frobenius on group representations started with the motivation of finding analogous factorisations of the group determinants for any finite "G"; that is, the determinants of arbitrary matrices representing elements of "K"["G"] acting by multiplication on the basis elements given by "g" in "G". Unless "G" is abelian, the factorisation must contain non-linear factors corresponding to irreducible representations of "G" of degree > 1.
Topological group case.
For a topological group "G", the regular representation in the above sense should be replaced by a suitable space of functions on "G", with "G" acting by translation. See Peter–Weyl theorem for the compact case. If "G" is a Lie group but not compact nor abelian, this is a difficult matter of harmonic analysis. The locally compact abelian case is part of the Pontryagin duality theory.
Normal bases in Galois theory.
In Galois theory it is shown that for a field "L", and a finite group "G" of automorphisms of "L", the fixed field "K" of "G" has ["L":"K"] = |"G"|. In fact we can say more: "L" viewed as a "K"["G"]-module is the regular representation. This is the content of the normal basis theorem, a normal basis being an element "x" of "L" such that the "g"("x") for "g" in "G" are a vector space basis for "L" over "K". Such "x" exist, and each one gives a "K"["G"]-isomorphism from "L" to "K"["G"]. From the point of view of algebraic number theory it is of interest to study "normal integral bases", where we try to replace "L" and "K" by the rings of algebraic integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: "a" + "bi" and "a" − "bi" can never form a Z-module basis of Z["i"] because 1 cannot be an integer combination. The reasons are studied in depth in Galois module theory.
More general algebras.
The regular representation of a group ring is such that the left-hand and right-hand regular representations give isomorphic modules (and we often need not distinguish the cases). Given an algebra over a field "A", it doesn't immediately make sense to ask about the relation between "A" as left-module over itself, and as right-module. In the group case, the mapping on basis elements "g" of "K"["G"] defined by taking the inverse element gives an isomorphism of "K"["G"] to its "opposite" ring. For "A" general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been shown to be related to topological quantum field theory in 1 + 1 dimensions by a particular instance of the cobordism hypothesis.


Big q-Laguerre polynomials

In mathematics, the big "q"-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
formula_1
Relation to other polynomials.
Big q-Laguerre polynomials→Laguerre polynomials


ADE classification

In mathematics, the ADE classification (originally "A-D-E" classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in . The complete list of simply laced Dynkin diagrams comprises
Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of formula_2 (no edge between the vertices) or formula_3 (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting formula_4 and formula_5), and three of the five exceptional Dynkin diagrams (omitting formula_6 and formula_7).
This list is non-redundant if one takes formula_8 for formula_9 If one extends the families to include redundant terms, one obtains the exceptional isomorphisms
and corresponding isomorphisms of classified objects.
The "A", "D", "E" nomenclature also yields the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.
Lie algebras.
In terms of complex semisimple Lie algebras:
In terms of compact Lie algebras and corresponding simply laced Lie groups:
Binary polyhedral groups.
The same classification applies to discrete subgroups of formula_23, the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced "affine" Dynkin diagrams formula_24 and the representations of these groups can be understood in terms of these diagrams. This connection is known as the after John McKay. The connection to Platonic solids is described in . The correspondence uses the construction of McKay graph.
Note that the ADE correspondence is "not" the correspondence of Platonic solids to their reflection group of symmetries: for instance, in the ADE correspondence the tetrahedron, cube/octahedron, and dodecahedron/icosahedron correspond to formula_25 while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the Coxeter groups formula_26 and formula_27
The orbifold of formula_28 constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a du Val singularity.
The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a "pair" of binary polyhedral groups. This is known as the Slodowy correspondence, named after Peter Slodowy – see .
Labeled graphs.
The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties, which can be stated in terms of the discrete Laplace operators or Cartan matrices. Proofs in terms of Cartan matrices may be found in .
The affine ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property:
That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent vertices minus value of vertex) – the positive solutions to the homogeneous equation:
Equivalently, the positive functions in the kernel of formula_30 The resulting numbering is unique up to scale, and if normalized such that the smallest number is 1, consists of small integers – 1 through 6, depending on the graph.
The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:
In terms of the Laplacian, the positive solutions to the inhomogeneous equation:
The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E8 they range from 58 to 270, and have been observed as early as .
Other classifications.
The elementary catastrophes are also classified by the ADE classification.
The ADE diagrams are exactly the quivers of finite type, via Gabriel's theorem.
There is also a link with generalized quadrangles, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems "E"6, "E"7 and "E"8.
The classes "A" and "D" correspond degenerate cases where the line set is empty or we have all lines passing through a fixed point, respectively.
There are deep connections between these objects, hinted at by the classification; some of these connections can be understood via string theory and quantum mechanics.
It was suggested that symmetries of small droplet clusters may be subject to an ADE classification.
The minimal models of two-dimensional conformal field theory have an ADE classification.
Four dimensional formula_32 superconformal gauge quiver theories with unitary gauge groups have an ADE classification.
Trinities.
Arnold has subsequently proposed many further connections in this vein, under the rubric of "mathematical trinities", and McKay has extended his correspondence along parallel and sometimes overlapping lines. Arnold terms these "trinities" to evoke religion, and suggest that (currently) these parallels rely more on faith than on rigorous proof, though some parallels are elaborated. Further trinities have been suggested by other authors. Arnold's trinities begin with R/C/H (the real numbers, complex numbers, and quaternions), which he remarks "everyone knows", and proceeds to imagine the other trinities as "complexifications" and "quaternionifications" of classical (real) mathematics, by analogy with finding symplectic analogs of classic Riemannian geometry, which he had previously proposed in the 1970s. In addition to examples from differential topology (such as characteristic classes), Arnold considers the three Platonic symmetries (tetrahedral, octahedral, icosahedral) as corresponding to the reals, complexes, and quaternions, which then connects with McKay's more algebraic correspondences, below.
McKay's correspondences are easier to describe. Firstly, the extended Dynkin diagrams formula_33 (corresponding to tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups formula_34 respectively, and the associated foldings are the diagrams formula_35 (note that in less careful writing, the extended (tilde) qualifier is often omitted). More significantly, McKay suggests a correspondence between the nodes of the formula_36 diagram and certain conjugacy classes of the monster group, which is known as "McKay's E8 observation;" see also monstrous moonshine. McKay further relates the nodes of formula_37 to conjugacy classes in 2."B" (an order 2 extension of the baby monster group), and the nodes of formula_38 to conjugacy classes in 3."Fi"24' (an order 3 extension of the Fischer group) – note that these are the three largest sporadic groups, and that the order of the extension corresponds to the symmetries of the diagram.
Turning from large simple groups to small ones, the corresponding Platonic groups formula_39 have connections with the projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660), which is deemed a "McKay correspondence". These groups are the only (simple) values for "p" such that PSL(2,"p") acts non-trivially on "p" points, a fact dating back to Évariste Galois in the 1830s. In fact, the groups decompose as products of sets (not as products of groups) as: formula_40 formula_41 and formula_42 These groups also are related to various geometries, which dates to Felix Klein in the 1870s; see icosahedral symmetry: related geometries for historical discussion and for more recent exposition. Associated geometries (tilings on Riemann surfaces) in which the action on "p" points can be seen are as follows: PSL(2,5) is the symmetries of the icosahedron (genus 0) with the compound of five tetrahedra as a 5-element set, PSL(2,7) of the Klein quartic (genus 3) with an embedded (complementary) Fano plane as a 7-element set (order 2 biplane), and PSL(2,11) the (genus 70) with embedded Paley biplane as an 11-element set (order 3 biplane). Of these, the icosahedron dates to antiquity, the Klein quartic to Klein in the 1870s, and the buckyball surface to Pablo Martin and David Singerman in 2008.
Algebro-geometrically, McKay also associates E6, E7, E8 respectively with: the 27 lines on a cubic surface, the 28 bitangents of a plane quartic curve, and the 120 tritangent planes of a canonic sextic curve of genus 4. The first of these is well-known, while the second is connected as follows: projecting the cubic from any point not on a line yields a double cover of the plane, branched along a quartic curve, with the 27 lines mapping to 27 of the 28 bitangents, and the 28th line is the image of the exceptional curve of the blowup. Note that the fundamental representations of E6, E7, E8 have dimensions 27, 56 (28·2), and 248 (120+128), while the number of roots is 27+45 = 72, 56+70 = 126, and 112+128 = 240.
This should also fit into the scheme of relating E8,7,6 with the largest three of the Sporadic simple groups, Monster, Baby and Fischer 24', cf. Monstrous Moonshine.