Real element

In group theory, a discipline within modern algebra, an element formula_1 of a group formula_2 is called a real element of formula_2 if it belongs to the same conjugacy class as its inverse formula_4, that is, if there is a formula_5 in formula_2 with formula_7, where formula_8 is defined as formula_9. An element formula_1 of a group formula_2 is called strongly real if there is an involution formula_12 with formula_13.
An element formula_1 of a group formula_2 is real if and only if for all representations formula_16 of formula_2, the trace formula_18 of the corresponding matrix is a real number. In other words, an element formula_1 of a group formula_2 is real if and only if formula_21 is a real number for all characters formula_22 of formula_2.
A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group formula_24 of any degree formula_25 is ambivalent.
Properties.
A group with real elements other than the identity element necessarily is of even order.
For a real element formula_1 of a group formula_2, the number of group elements formula_5 with formula_7 is equal to formula_30, where formula_31 is the centralizer of formula_1,
Every involution is strongly real. Furthermore, every element that is the product of two involutions is strongly real. Conversely, every strongly real element is the product of two involutions.
If and formula_1 is real in formula_2 and formula_30 is odd, then formula_1 is strongly real in formula_2.
Extended centralizer.
The extended centralizer of an element formula_1 of a group formula_2 is defined as
making the extended centralizer of an element formula_1 equal to the normalizer of the set
The extended centralizer of an element of a group formula_2 is always a subgroup of formula_2. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element formula_1 of a group formula_2 that is not an involution,


Reduced ring

In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, "x"2 = 0 implies "x" = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.
The nilpotent elements of a commutative ring "R" form an ideal of "R", called the nilradical of "R"; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero.
A quotient ring "R/I" is reduced if and only if "I" is a radical ideal.
Let "D" be the set of all zerodivisors in a reduced ring "R". Then "D" is the union of all minimal prime ideals.
Over a Noetherian ring "R", we say a finitely generated module "M" has locally constant rank if formula_1 is a locally constant (or equivalently continuous) function on Spec "R". Then "R" is reduced if and only if every finitely generated module of locally constant rank is projective.
Generalizations.
Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of a reduced scheme.


Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.
Classes.
Several important classes of groups are algebraic groups, including:
There are other algebraic groups, but Chevalley's structure theorem asserts that every algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if "K" is a perfect field, and "G" an algebraic group over "K", there exists a unique normal closed subgroup "H" in "G", such that "H" is a linear algebraic group and "G"/"H" an abelian variety.
According to another basic theorem, any group that is also an affine variety has a faithful finite-dimensional linear representation: it is isomorphic to a matrix group, defined by polynomial equations.
Over the fields of real and complex numbers, every algebraic group is also a Lie group, but the converse is false
A group scheme is a generalization of an algebraic group that allows, in particular, working over a commutative ring instead of a field.
Algebraic subgroup.
An algebraic subgroup of an algebraic group is a Zariski-closed subgroup.
Generally these are taken to be connected (or irreducible as a variety) as well.
Another way of expressing the condition is as a subgroup that is also a subvariety.
This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic "p".
Coxeter groups.
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is formula_1, and the number of elements of the general linear group over a finite field is the "q"-factorial formula_2; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
Glossary of algebraic groups.
There are a number of mathematical notions to study and classify algebraic groups.
In the sequel, "G" denotes an algebraic group over a field "k".


Rahul Pandharipande

Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry. His particular interests
concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas invariants, and the cohomology of the moduli space of curves. His father Vijay Raghunath Pandharipande was a renowned theoretical physicist who worked in the area of nuclear physics.
Educational and professional history.
He received his A.B. from Princeton University in 1990 and his Ph.D. from Harvard University in 1994 with a thesis entitled `"A Compactification over the Moduli Space of Stable Curves of the Universal Moduli Space of Slope-Semistable Vector Bundles"'. His thesis advisor at Harvard was Joe Harris. After teaching at the University of Chicago and the California Institute of Technology, he joined the faculty as Professor of Mathematics at Princeton University in 2002. In 2011, he accepted a Professorship at ETH Zürich.


List of algebraic geometry topics

This is a list of algebraic geometry topics, by Wikipedia page.


Unirationality


List of number fields with class number one

This is an incomplete list of number fields with class number 1.
It is believed that there are infinitely many such number fields, but this has not been proven.
Definition.
The class number of a number field is by definition the order of the ideal class group of its ring of integers.
Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
Quadratic number fields.
These are of the form "K" = Q(), for a square-free integer "d".
Real quadratic fields.
"K" is called real quadratic if "d" > 0. "K" has class number 1 for the following values of "d" :
Despite what would appear to be the case for these small values, not all prime numbers that are congruent to 1 modulo 4 appear on this list, notably the fields Q() for "d" = 229 and "d" = 257 both have class number greater than 1 (in fact equal to 3 in both cases). The density of such primes for which Q() does have class number 1 is conjectured to be nonzero, and in fact close to 76%,
however it is not even known whether there are infinitely many real quadratic fields with class number 1.
Imaginary quadratic fields.
"K" has class number 1 exactly for the following negative values of "d":
Cubic fields.
Totally real cubic field.
The first 60 totally real cubic fields (ordered by discriminant) have class number one. In other words, all cubic fields of discriminant between 0 and 1944 (inclusively) have class number one. The next totally real cubic field (of discriminant 1957) has class number two. The polynomials defining the totally real cubic fields that have discriminants less than 500 with class number one are:
Complex cubic field.
All complex cubic fields with discriminant greater than −500 have class number one, except the fields with discriminants −283, −331 and −491 which have class number 2. The polynomials defining the complex cubic fields that have class number one and discriminant greater than −500 are:
Cyclotomic fields.
The following is a complete list of "n" for which the field Q(ζ"n") has class number 1:
On the other hand, the maximal real subfields Q(cos(2π/2"n")) of the 2-power cyclotomic fields Q(ζ2"n") (where "n" is a positive integer) are known to have class number 1 for n≤8, and
it is conjectured that they have class number 1 for all "n". Weber showed that these fields have odd class number. In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 107, and later improved this bound to 109. These fields are the "n"-th layers of the cyclotomic Z2-extension of Q. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic Z3-extension of Q have no prime factor less than 104. Coates has raised the question of whether, for all primes "p", every layer of the cyclotomic Z"p"-extension of Q has class number 1.
CM fields.
Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field "K", i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1. He showed that there are finitely many of a fixed degree. Shortly thereafter, Andrew Odlyzko showed that there are only finitely many "Galois" CM fields of class number 1. In 2001, V. Kumar Murty showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1.
A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject. Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.


Moore determinant of a Hermitian matrix

In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by .


Unipotent representation

In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups.
Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group.
Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense.
Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction.
Finite fields.
Over finite fields, the unipotent representations are those that occur in the decomposition of the Deligne–Lusztig characters "R" of the trivial representation 1 of a torus "T" . They were classified by .
Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the Steinberg representation, and θ10.
Non-archimedean local fields.
 classified the unipotent characters over non-archimedean local fields.
Archimedean local fields.
 discusses several different possible definitions of unipotent representations of real Lie groups.


Peter Johnstone (mathematician)

Peter Tennant Johnstone (born 1948) is Professor of the Foundations of Mathematics at the University of Cambridge, and a fellow of St. John's College.
He invented or developed a broad range of fundamental ideas in topos theory. His thesis, completed at the University of Cambridge in 1974, was entitled "Some Aspects of Internal Category Theory in an Elementary Topos".
He is a great-great nephew of the Reverend George Gilfillan who was apostrophised in William McGonagall's first poem.


Zoghman Mebkhout

Zoghman Mebkhout (born 1949 ) (مبخوت زغمان) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of "D"-modules.
Career.
Mebkhout is currently a research director at the French National Centre for Scientific Research and in 2002 Zoghman received the Servant Medal from the CNRS a prize given every two years with an amount of €10,000.
Notable works.
In September 1979 Mebkhout presented the Riemann–Hilbert correspondence, which is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups.
In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. Certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions correspond. An independent proof of this result was presented by Masaki Kashiwara in April 1980.
Zoghman is now largely known as a specialist in "D"-modules theory.
Recognition.
Zoghman is one of the first modern international-caliber North-African mathematicians. A symposium in Spain was held on his sixtieth birthday. He was invited to the Institute for Advanced Study and gave a recent talk at Institut Fourier.
Alexander Grothendieck said that Mebkhout's name was hidden and his role neglected in an operation headed by Pierre Deligne in the Luminy congress in June 1981. He calls it "a great disgrace of the mathematical world of this century" and is one of the reasons of Grothendieck's departure from mathematics.


Duflo isomorphism

In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by .
The isomorphism also follows from the Kontsevich formality theorem.
Properties.
For a nilpotent Lie algebra the Duflo isomorphism coincides with the symmetrization map from symmetric algebra to universal enveloping algebra. For a semisimple Lie algebra the Duflo isomorphism is compatible in a natural way with the Harish-Chandra isomorphism.


Monster group

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
   2463205976112133171923293141475971
= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8.
The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the "happy family", and the remaining six exceptions "pariahs".
It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in "Scientific American".
History.
The monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. Griess constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra over the real numbers; he first announced his construction in Ann Arbor on January 14, 1980. In his 1982 paper, he referred to the monster as the Friendly Giant, but this name has not been generally adopted. John Conway and Jacques Tits subsequently simplified this construction.
Griess's construction showed that the monster exists. Thompson showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a 196,883-dimensional faithful representation. A proof of the existence of such a representation was announced by Norton, though he has never published the details. Griess, Meierfrankenfeld and Segev gave the first complete published proof of the uniqueness of the monster (more precisely, they showed that a group with the same centralizers of involutions as the monster is isomorphic to the monster).
The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: the Fischer group Fi24, the baby monster, and the Conway group Co1.
The Schur multiplier and the outer automorphism group of the monster are both trivial.
Representations.
The minimal degree of a faithful complex representation is 196,883, which is the product of the three largest prime divisors of the order of M.
The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation.
The smallest faithful permutation representation of the monster is on
24 · 37 · 53 · 74 · 11 · 132 · 29 · 41 · 59 · 71 (about 1020)
points.
The monster can be realized as a Galois group over the rational numbers, and as a Hurwitz group.
The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A100 and SL20(2) are far larger, but easy to calculate with as they have "small" permutation or linear representations. The alternating groups have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension 4370).
A computer construction.
Robert A. Wilson has found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.
Wilson asserts that the best description of the monster is to say, "It is the automorphism group of the monster vertex algebra". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".
Wilson with collaborators has found a method of performing calculations with the monster that is considerably faster. Let "V" be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup "H" (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup "H" chosen is 31+12.2.Suz.2, where Suz is the Suzuki group. Elements of the monster are stored as words in the elements of "H" and an extra generator "T". It is reasonably quick to calculate the action of one of these words on a vector in "V". Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors "u" and "v" whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element "g" of the monster by finding the smallest "i" > 0 such that "g""i""u" = "u" and "g""i""v" = "v".
This and similar constructions (in different characteristics) have been used to find some of its non-local maximal subgroups.
Moonshine.
The monster group is one of two principal constituents in the monstrous moonshine conjecture by Conway and Norton, which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992.
In this setting, the monster group is visible as the automorphism group of the monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the monster Lie algebra, a generalized Kac–Moody algebra.
Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object. Conway said of the monster group: "There's never been any kind of explanation of why it's there, and it's obviously not there just by coincidence. It's got too many intriguing properties for it all to be just an accident." Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God."
McKay's E8 observation.
There are also connections between the monster and the extended Dynkin diagrams formula_1 specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as "McKay's E8 observation". This is then extended to a relation between the extended diagrams formula_2 and the groups 3.Fi24′, 2.B, and M, where these are (3/2/1-fold central extensions) of the Fischer group, baby monster group, and monster. These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the monster, and the order of the extension corresponds to the symmetries of the diagram. See ADE classification: trinities for further connections (of McKay correspondence type), including (for the monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known as Bring's curve.
Maximal subgroups.
The monster has at least 44 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A12.
The monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book "Symmetry and the Monster" by Mark Ronan, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.
Forty-four of the classes of maximal subgroups of the monster are given by the following list, which is (as of 2016) believed to be complete except possibly for almost simple subgroups with non-abelian simple socles of the form L2(13), U3(4), or U3(8). However, tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups on the list below were incorrectly omitted from some previous lists.
Sources.
Footnotes


CH-quasigroup

In mathematics, a CH-quasigroup, introduced by , is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH" stands for cubic hypersurface.


Double coset

In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in , the -double coset of is the set
When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation
The set of all double cosets is denoted by formula_2
Properties.
Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.
There is an equivalent description of double cosets in terms of single cosets. Let and both act by right multiplication on . Then acts by left multiplication on the product of coset spaces . The orbits of this action are in one-to-one correspondence with . This correspondence identifies with the double coset . Briefly, this is because every -orbit admits representatives of the form , and the representative is determined only up to left multiplication by an element of . Similarly, acts by right multiplication on , and the orbits of this action are in one-to-one correspondence with the double cosets . Conceptually, this identifies the double coset space with the space of relative configurations of an -coset and a -coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups , the space of -multicosets is the set of -orbits of .
The analog of Lagrange's theorem for double cosets is false. This means that the size of a double coset need not divide the order of . For example, let be the symmetric group on three letters, and let and be the cyclic subgroups generated by the transpositions and , respectively. If denotes the identity permutation, then
This has four elements, and four does not divide six, the order of . It is also false that different double cosets have the same size. Continuing the same example,
which has two elements, not four.
However, suppose that is normal. As noted earlier, in this case the double coset space equals the left coset space . Similarly, if is normal, then is the right coset space . Standard results about left and right coset spaces then imply the following facts.
Products in the free abelian group on the set of double cosets.
Suppose that is a group and that , , and are subgroups. Under certain finiteness conditions, there is a product on the free abelian group generated by the - and -double cosets with values in the free abelian group generated by the -double cosets. This means there is a bilinear function
Assume for simplicity that is finite. To define the product, reinterpret these free abelian groups in terms of the group algebra of as follows. Every element of has the form
where is a set of integers indexed by the elements of . This element may be interpreted as a -valued function on , specifically, . This function may be pulled back along the projection which sends to the double coset . This results in a function . By the way in which this function was constructed, it is left invariant under and right invariant under . The corresponding element of the group algebra is
and this element is invariant under left multiplication by and right multiplication by . Conceptually, this element is obtained by replacing by the elements it contains, and the finiteness of ensures that the sum is still finite. Conversely, every element of which is left invariant under and right invariant under is the pullback of a function on . Parallel statements are true for and .
When elements of , , and are interpreted as invariant elements of , then the product whose existence was asserted above is precisely the multiplication in . Indeed, it is trivial to check that the product of a left--invariant element and a right--invariant element continues to be left--invariant and right--invariant. The bilinearity of the product follows immediately from the bilinearity of multiplication in . It also follows that if is a fourth subgroup of , then the product of -, -, and -double cosets is associative. Because the product in corresponds to convolution of functions on , this product is sometimes called the convolution product.
An important special case is when . In this case, the product is a bilinear function
This product turns into an associative ring whose identity element is the class of the trivial double coset . In general, this ring is non-commutative. For example, if , then the ring is the group algebra , and a group algebra is a commutative ring if and only if the underlying group is abelian.
If is normal, so that the -double cosets are the same as the elements of the quotient group , then the product on is the product in the group algebra . In particular, it is the usual convolution of functions on . In this case, the ring is commutative if and only if is abelian, or equivalently, if and only if contains the commutator subgroup of .
If is not normal, then may be commutative even if is non-abelian. A classical example is the product of two Hecke operators. This is the product in the Hecke algebra, which is commutative even though the group is the modular group, which is non-abelian, and the subgroup is an arithmetic subgroup and in particular does not contain the commutator subgroup. Commutativity of the convolution product is closely tied to Gelfand pairs.
When the group is a topological group, it is possible to weaken the assumption that the number of left and right cosets in each double coset is finite. The group algebra is replaced by an algebra of functions such as or , and the sums are replaced by integrals. The product still corresponds to convolution. For instance, this happens for the Hecke algebra of a locally compact group.
Applications.
When a group formula_18 has a transitive group action on a set formula_19, computing certain double coset decompositions of formula_18 reveals extra information about structure of the action of formula_18 on formula_22. Specifically, if formula_23 is the stabilizer subgroup of some element formula_24, then formula_18 decomposes as exactly two double cosets of formula_26 if and only if formula_18 acts transitively on the set of distinct pairs of formula_19. See 2-transitive groups for more information about this action.
Double cosets are important in connection with representation theory, when a representation of is used to construct an induced representation of , which is then restricted to . The corresponding double coset structure carries information about how the resulting representation decomposes. In the case of finite groups, this is Mackey's decomposition theorem.
They are also important in functional analysis, where in some important cases functions left-invariant and right-invariant by a subgroup can form a commutative ring under convolution: see Gelfand pair.
In geometry, a Clifford–Klein form is a double coset space , where is a reductive Lie group, is a closed subgroup, and is a discrete subgroup (of ) that acts properly discontinuously on the homogeneous space .
In number theory, the Hecke algebra corresponding to a congruence subgroup of the modular group is spanned by elements of the double coset space formula_29; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators formula_30 corresponding to the double cosets formula_31 or formula_32, where formula_33 (these have different properties depending on whether and are coprime or not), and the diamond operators formula_34 given by the double cosets formula_35 where formula_36 and we require formula_37 (the choice of does not affect the answer).


Galois module

In mathematics, a Galois module is a "G"-module, with "G" being the Galois group of some extension of fields. The term Galois representation is frequently used when the "G"-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for "G"-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
Examples.
Ramification theory.
Let "K" be a valued field (with valuation denoted "v") and let "L"/"K" be a finite Galois extension with Galois group "G". For an extension "w" of "v" to "L", let "Iw" denote its inertia group. A Galois module ρ : "G" → Aut("V") is said to be unramified if ρ("Iw") = {1}.
Galois module structure of algebraic integers.
In classical algebraic number theory, let "L" be a Galois extension of a field "K", and let "G" be the corresponding Galois group. Then the ring "O""L" of algebraic integers of "L" can be considered as an "O""K"["G"]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that "L" is a free "K"["G"]-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e. of α in "O""L" such that its conjugate elements under "G" give a free basis for "O""L" over "O""K". This is an interesting question even (perhaps especially) when "K" is the rational number field Q.
For example, if "L" = Q(), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q("ζ") where
In fact all the subfields of the cyclotomic fields for "p"-th roots of unity for "p" a "prime number" have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand, the Gaussian field does not. This is an example of a "necessary" condition found by Emmy Noether ("perhaps known earlier?"). What matters here is "tame" ramification. In terms of the discriminant "D" of "L", and taking still "K" = Q, no prime "p" must divide "D" to the power "p". Then Noether's theorem states that tame ramification is necessary and sufficient for "OL" to be a projective module over Z["G"]. It is certainly therefore necessary for it to be a "free" module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.
A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomic field.
Galois representations in number theory.
Many objects that arise in number theory are naturally Galois representations. For example, if "L" is a Galois extension of a number field "K", the ring of integers "OL" of "L" is a Galois module over "OK" for the Galois group of "L"/"K" (see Hilbert–Speiser theorem). If "K" is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of "K" and its study leads to local class field theory. For global class field theory, the union of the idele class groups of all finite separable extensions of "K" is used instead.
There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the ℓ-adic Tate modules of abelian varieties.
Artin representations.
Let "K" be a number field. Emil Artin introduced a class of Galois representations of the absolute Galois group "GK" of "K", now called Artin representations. These are the continuous finite-dimensional linear representations of "GK" on complex vector spaces. Artin's study of these representations led him to formulate the Artin reciprocity law and conjecture what is now called the Artin conjecture concerning the holomorphy of Artin "L"-functions.
Because of the incompatibility of the profinite topology on "GK" and the usual (Euclidean) topology on complex vector spaces, the image of an Artin representation is always finite.
ℓ-adic representations.
Let ℓ be a prime number. An ℓ-adic representation of "GK" is a continuous group homomorphism where "M" is either a finite-dimensional vector space over ℓ (the algebraic closure of the ℓ-adic numbers Qℓ) or a finitely generated ℓ-module (where ℓ is the integral closure of Zℓ in ℓ). The first examples to arise were the ℓ-adic cyclotomic character and the ℓ-adic Tate modules of abelian varieties over "K". Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on ℓ-adic cohomology groups of algebraic varieties.
Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of "G"Q under the ℓ-adic cyclotomic character is formula_1. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of ℓ with C they can be identified with "bona fide" Artin representations.
Mod ℓ representations.
These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.
Local conditions on representations.
There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include:
Representations of the Weil group.
If "K" is a local or global field, the theory of class formations attaches to "K" its Weil group "WK", a continuous group homomorphism , and an isomorphism of topological groups
where "CK" is "K"× or the idele class group "IK"/"K"× (depending on whether "K" is local or global) and is the abelianization of the Weil group of "K". Via φ, any representation of "GK" can be considered as a representation of "WK". However, "WK" can have strictly more representations than "GK". For example, via "rK" the continuous complex characters of "WK" are in bijection with those of "CK". Thus, the absolute value character on "CK" yields a character of "WK" whose image is infinite and therefore is not a character of "GK" (as all such have finite image).
An ℓ-adic representation of "WK" is defined in the same way as for "GK". These arise naturally from geometry: if "X" is a smooth projective variety over "K", then the ℓ-adic cohomology of the geometric fibre of "X" is an ℓ-adic representation of "GK" which, via φ, induces an ℓ-adic representation of "WK". If "K" is a local field of residue characteristic "p" ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of "WK".
Weil–Deligne representations.
Let "K" be a local field. Let "E" be a field of characteristic zero. A Weil–Deligne representation over "E" of "WK" (or simply of "K") is a pair ("r", "N") consisting of
These representations are the same as the representations over "E" of the Weil–Deligne group of "K".
If the residue characteristic of "K" is different from ℓ, Grothendieck's ℓ-adic monodromy theorem sets up a bijection between ℓ-adic representations of "WK" (over ℓ) and Weil–Deligne representations of "WK" over ℓ (or equivalently over C). These latter have the nice feature that the continuity of "r" is only with respect to the discrete topology on "V", thus making the situation more algebraic in flavor.


Day convolution

In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal category structure on the category of functors formula_1 over some monoidal category formula_2.
Definition.
Let formula_3 be a monoidal category enriched over a symmetric monoidal closed category formula_4. Given two functors formula_5, we define their Day convolution as the following coend.
If formula_7 is symmetric, then formula_8 is also symmetric. We can show this defines an associative monoidal product.


Trivial group

In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1 or "e" depending on the context. If the group operation is denoted ∗ then it is defined by .
The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.
The trivial group should not be confused with the empty set (which has no elements, and lacking an identity element, cannot be a group).
Given any group "G", the group consisting of only the identity element is a subgroup of "G", and, being the trivial group, is called the trivial subgroup of "G".
The term, when referred to ""G" has no nontrivial proper subgroups" refers to the only subgroups of "G" being the trivial group {"e"} and the group "G" itself.
Properties.
The trivial group is cyclic of order 1; as such it may be denoted Z1 or C1. If the group operation is called addition, the trivial group is usually denoted by 0. If the group operation is called multiplication then 1 can be a notation for the trivial group. (Combining these leads to the trivial ring in which the addition and multiplication operations are identical and 0=1.)
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
The trivial group can be made a (bi-)ordered group by equipping it with the trivial non-strict order ≤.


Root datum

In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.
Definition.
A root datum consists of a quadruple
where
The elements of formula_5 are called the roots of the root datum, and the elements of formula_7 are called the coroots.
If formula_5 does not contain formula_24 for any formula_25, then the root datum is called reduced.
The root datum of an algebraic group.
If formula_26 is a reductive algebraic group over an algebraically closed field formula_27 with a split maximal torus formula_28 then its root datum is a quadruple
where
A connected split reductive algebraic group over formula_27 is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.
For any root datum formula_29, we can define a dual root datum formula_36 by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If formula_26 is a connected reductive algebraic group over the algebraically closed field formula_27, then its Langlands dual group formula_39 is the complex connected reductive group whose root datum is dual to that of formula_26.


Unique factorization domain

In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units.
Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.
Unique factorization domains appear in the following chain of class inclusions:
Definition.
Formally, a unique factorization domain is defined to be an integral domain "R" in which every non-zero element "x" of "R" can be written as a product (an empty product if "x" is a unit) of irreducible elements "p"i of "R" and a unit "u":
and this representation is unique in the following sense:
If "q"1, ..., "q""m" are irreducible elements of "R" and "w" is a unit such that
then "m" = "n", and there exists a bijective map "φ" : {1, ..., "n"} → {1, ..., "m"} such that "p""i" is associated to "q""φ"("i") for "i" ∈ {1, ..., "n"}.
The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:
Examples.
Most rings familiar from elementary mathematics are UFDs:
Properties.
Some concepts defined for integers can be generalized to UFDs:
Equivalent conditions for a ring to be a UFD.
A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain.
In general, for an integral domain "A", the following conditions are equivalent:
In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID.
For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains height one prime ideal (induction on height) which is principal. By (2), the ring is a UFD.


Base change lifting

In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup.
The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifting was introduced by for Hilbert modular forms of cyclic totally real fields of prime degree, by comparing the trace of twisted Hecke operators on Hilbert modular forms with the trace of Hecke operators on ordinary modular forms. gave a representation theoretic interpretation of Saito's results and used this to generalize them. extended the base change lifting to more general automorphic forms and showed how to use the base change lifting for GL2 to prove the Artin conjecture for tetrahedral and some octahedral 2-dimensional representations of the Galois group.
, and gave expositions of the base change lifting for GL2 and its applications to the Artin conjecture.
Properties.
If "E"/"F" is a finite cyclic Galois extension of global fields, then the base change lifting of gives a map from automorphic forms for GL"n"("F") to automorphic forms for GL"n"("E") = Res"E"/"F"GL"n"("F"). This base change lifting is the special case of Langlands functoriality, corresponding (roughly) to the diagonal embedding of the Langlands dual GL"n"(C) of GL"n" to the Langlands dual GL"n"(C)×...×GL"n"(C) of Res"E"/"F"GL"n".


Coproduct

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
Definition.
Let "C" be a category and let "X"1 and "X"2 be objects of "C". An object is called the coproduct of "X"1 and "X"2, written "X"1 ∐ "X"2 or "X"1 ⊕ "X"2, or sometimes simply "X"1 + "X"2, if there exist morphisms "i"1 : "X"1 → "X"1 ∐ "X"2 and "i"2 : "X"2 → "X"1 ∐ "X"2 satisfying the following universal property: for any object "Y" and any morphisms "f"1 : "X"1 → Y and "f"2 : "X"2 → Y, there exists a unique morphism "f" : "X"1 ∐ "X"2 → "Y" such that "f"1 = "f" ∘ "i"1 and "f"2 = "f" ∘ "i"2. That is, the following diagram commutes:
The unique arrow "f" making this diagram commute may be denoted "f"1 ∐ "f"2, "f"1 ⊕ "f"2, "f"1 + "f"2 or ["f"1, "f"2]. The morphisms "i"1 and "i"2 are called "canonical injections", although they need not be injections or even monic.
The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set "J". The coproduct of the family {"Xj" : "j" ∈ "J"} is an object "X" together with a collection of morphisms "ij" : "Xj" → "X" such that, for any object "Y" and any collection of morphisms "fj" : "Xj" → "Y", there exists a unique morphism "f" from "X" to "Y" such that "fj" = "f" ∘ "ij". That is, the following diagram commutes for each "j" in "J":
The coproduct "X" of the family {"Xj"} is often denoted formula_1 or formula_2
Sometimes the morphism "f: X → Y" may be denoted formula_3 to indicate its dependence on the individual "f""j"s.
Examples.
The coproduct in the category of sets is simply the disjoint union with the maps "ij" being the inclusion maps. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)
Given a commutative ring "R", the coproduct in the category of commutative "R"-algebras is the tensor product. In the category of (noncommutative) "R"-algebras, the coproduct is a quotient of the tensor algebra (see free product of associative algebras).
In the case of topological spaces coproducts are disjoint unions with their disjoint union topologies. That is, it is a disjoint union of the underlying sets, and the open sets are sets "open in each of the spaces", in a rather evident sense. In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
Despite all this dissimilarity, there is still, at the heart of the whole thing, a disjoint union: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.
The coproduct of a poset category is the join operation.
Discussion.
The coproduct construction given above is actually a special case of a colimit in category theory. The coproduct in a category formula_4 can be defined as the colimit of any functor from a discrete category formula_5 into formula_4. Not every family formula_7 will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if formula_8 and formula_9 are two coproducts of the family formula_7, then (by the definition of coproducts) there exists a unique isomorphism formula_11 such that formula_12 for each formula_13.
As with any universal property, the coproduct can be understood as a universal morphism. Let formula_14 be the diagonal functor which assigns to each object formula_15 the ordered pair formula_16 and to each morphism formula_17 the pair formula_18. Then the coproduct formula_19 in formula_4 is given by a universal morphism to the functor formula_21 from the object formula_22 in formula_23.
The coproduct indexed by the empty set (that is, an "empty coproduct") is the same as an initial object in formula_4.
If formula_5 is a set such that all coproducts for families indexed with formula_5 exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor formula_27. The coproduct of the family formula_7 is then often denoted by
and the maps formula_30 are known as the natural injections.
Letting formula_31 denote the set of all morphisms from formula_32 to formula_33 in formula_4 (that is, a hom-set in formula_4), we have a natural isomorphism
given by the bijection which maps every tuple of morphisms
(a product in Set, the category of sets, which is the Cartesian product, so it is a tuple of morphisms) to the morphism
That this map is a surjection follows from the commutativity of the diagram: any morphism formula_39 is the coproduct of the tuple
That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category formula_41 to Set is continuous; it preserves limits (a coproduct in formula_4 is a product in formula_41).
If formula_5 is a finite set, say formula_45, then the coproduct of objects formula_46 is often denoted by formula_47. Suppose all finite coproducts exist in "C", coproduct functors have been chosen as above, and 0 denotes the initial object of "C" corresponding to the empty coproduct. We then have natural isomorphisms
These properties are formally similar to those of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal category.
If the category has a zero object formula_51, then we have a unique morphism formula_52 (since formula_51 is terminal) and thus a morphism formula_54. Since formula_51 is also initial, we have a canonical isomorphism formula_56 as in the preceding paragraph. We thus have morphisms formula_57 and formula_58, by which we infer a canonical morphism formula_59. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in Grp it is a proper epimorphism while in Set* (the category of pointed sets) it is a proper monomorphism. In any preadditive category, this morphism is an isomorphism and the corresponding object is known as the biproduct. A category with all finite biproducts is known as a semiadditive category.
If all families of objects indexed by formula_5 have coproducts in formula_4, then the coproduct comprises a functor formula_27. Note that, like the product, this functor is "covariant".


Hartley transform

In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse.
The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983.
The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use.
Definition.
The Hartley transform of a function formula_1 is defined by:
where formula_3 can in applications be an angular frequency and
is the cosine-and-sine (cas) or "Hartley" kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).
Inverse transform.
The Hartley transform has the convenient property of being its own inverse (an involution):
Conventions.
The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties:
Relation to Fourier transform.
This transform differs from the classic Fourier transform
formula_14 in the choice of the kernel. In the Fourier transform, we have the exponential kernel:
formula_15
where formula_16 is the imaginary unit.
The two transforms are closely related, however, and the Fourier transform (assuming it uses the same formula_17 normalization convention) can be computed from the Hartley transform via:
That is, the real and imaginary parts of the Fourier transform are simply given by the even and odd parts of the Hartley transform, respectively.
Conversely, for real-valued functions "f"("t"), the Hartley transform is given from the Fourier transform's real and imaginary parts:
where formula_20 and formula_21 denote the real and imaginary parts of the complex Fourier transform.
Properties.
The Hartley transform is a real linear operator, and is symmetric (and Hermitian). From the symmetric and self-inverse properties, it follows that the transform is a unitary operator (indeed, orthogonal).
There is also an analogue of the convolution theorem for the Hartley transform. If two functions formula_22 and formula_23 have Hartley transforms formula_24 and formula_25, respectively, then their convolution formula_26 has the Hartley transform:
Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.
cas.
The properties of the "Hartley kernel", for which Hartley introduced the name "cas" for the function (from "cosine and sine") in 1942, follow directly from trigonometry, and its definition as a phase-shifted trigonometric function formula_28. For example, it has an angle-addition identity of:
Additionally:
and its derivative is given by:


Partial inverse of a matrix

In linear algebra and statistics, the partial inverse of a matrix is an operation related to Gaussian elimination which has applications in numerical analysis and statistics. It is also known by various authors as the principal pivot transform, or as the sweep, gyration, or exchange operator.
Given an formula_1 matrix formula_2 over a vector space formula_3 partitioned into blocks:
If formula_5 is invertible, then the partial inverse of formula_6 around the pivot block formula_5 is created by inverting formula_5, putting the Schur complement formula_9 in place of formula_10, and adjusting the off-diagonal elements accordingly:
Conceptually, partial inversion corresponds to a rotation of the graph of the matrix formula_12, such that, for conformally-partitioned column matrices formula_13 and formula_14:
As defined this way, this operator is its own inverse: formula_16, and if the pivot block formula_5 is chosen to be the entire matrix, then the transform simply gives the matrix inverse formula_18. Note that some authors define a related operation (under one of the other names) which is not an inverse per se; particularly, one common definition instead has formula_19.
The transform is often presented as a pivot around a single non-zero element formula_20, in which case one has
Partial inverses obey a number of nice properties:
Use of the partial inverse in numerical analysis is due to the fact that there is some flexibility in the choices of pivots, allowing for non-invertible elements to be avoided, and because the operation of "rotation" (of the graph of the pivoted matrix) has better numerical stability than the "shearing" operation which is implicitly performed by Gaussian elimination. Use in statistics is due to the fact that the resulting matrix nicely decomposes into blocks which have useful meanings in the context of linear regression.


Minuscule representation

In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group acts transitively on the weights. Some authors exclude the trivial representation. A quasi-minuscule representation (also called a basic representation) is an irreducible representation such that all non-zero weights are in the same orbit under the Weyl group; each simple Lie algebra has a unique quasi-minuscule representation that is not minuscule, and the multiplicity of the zero weight is the number of short nodes of the Dynkin diagram.
The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the dimensions of the minuscule representations are given as follows.


Convexity (algebraic geometry)

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces formula_1 in quantum cohomology.These moduli spaces are smooth orbifolds whenever the target space is convex. A variety formula_2 is called convex if the pullback of the tangent bundle to a stable rational curve formula_3 has globally generated sections. Geometrically this implies the curve is free to move around formula_2 infinitesimally without any obstruction. Convexity is generally phrased as the technical condition
since Serre's vanishing theorem guarantees this sheaf has globally generated sections. Intuitively this means that on a neighborhood of a point, with a vector field in that neighborhood, the local parallel transport can be extended globally. This generalizes the idea of convexity in Euclidean geometry, where given two points formula_6 in a convex set formula_7, all of the points formula_8 are contained in that set. There is a vector field formula_9 in a neighborhood formula_10 of formula_11 transporting formula_11 to each point formula_13. Since the vector bundle of formula_14 is trivial, hence globally generated, there is a vector field formula_15 on formula_14 such that the equality formula_17 holds on restriction.
Examples.
There are many examples of convex spaces, including the following.
Spaces with trivial rational curves.
If the only maps from a rational curve to formula_2 are constants maps, then the pullback of the tangent sheaf is the free sheaf formula_19 where formula_20. These sheaves have trivial non-zero cohomology, and hence they are always convex. In particular, Abelian varieties have this property since the Albanese variety of a rational curve formula_21 is trivial, and every map from a variety to an Abelian variety factors through the Albanese.
Projective spaces.
Projective spaces are examples of homogeneous spaces, but their convexity can also be proved using a sheaf cohomology computation. Recall the Euler sequence relates the tangent space through a short exact sequence
If we only need to consider degree formula_23 embeddings, there is a short exact sequence
giving the long exact sequence
since the first two formula_26-terms are zero, which follows from formula_21 being of genus formula_28, and the second calculation follows from the Riemann–Roch theorem, we have convexity of formula_29. Then, any nodal map can be reduced to this case by considering one of the components formula_30 of formula_21.
Homogeneous spaces.
Another large class of examples are homogenous spaces formula_32 where formula_33 is a parabolic subgroup of formula_34. These have globally generated sections since formula_34 acts transitively on formula_2, meaning it can take a bases in formula_37 to a basis in any other point formula_38, hence it has globally generated sections. Then, the pullback is always globally generated. This class of examples includes Grassmannians, projective spaces, and flag varieties.
Product spaces.
Also, products of convex spaces are still convex. This follows from the Kunneth theorem in coherent sheaf cohomology.
Applications.
There are many useful technical advantages of considering moduli spaces of stable curves on convex spaces. That is, the spaces formula_1 have nice geometric and deformation-theoretic properties.
Deformation theory.
The deformations of formula_3 in the Hilbert scheme of graphs formula_41 has tangent space
where formula_43 is the point in the scheme representing the map. Convexity of formula_2 gives the dimension formula below. In addition, convexity implies all infinitesimal deformations are unobstructed.
Structure.
These spaces are normal projective varieties of pure dimension
which are locally the quotient of a smooth variety by a finite group. Also, the open subvariety formula_46 parameterizing non-singular maps is a smooth fine moduli space. In particular, this implies the stacks formula_47 are orbifolds.
Boundary divisors.
The spaces formula_1 have nice boundary divisors given by
for a partition formula_50 of formula_51 and formula_52 the point lying along the intersection of two rational curves formula_53.


Dualizing module

In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.
Definition.
A dualizing module for a Noetherian ring "R" is a finitely generated module "M" such that for any maximal ideal "m", the "R"/"m" vector space vanishes if "n" ≠ height("m") and is 1-dimensional if "n" = height("m").
A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module.
In particular if the ring is local the dualizing module is unique up to isomorphism.
A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute.
Examples.
If "R" is a Gorenstein ring, then "R" considered as a module over itself is a dualizing module.
If "R" is an Artinian local ring then the Matlis module of "R" (the injective hull of the residue field) is the dualizing module.
The Artinian local ring "R" = "k"["x","y"]/("x"2,"y"2,"xy") has a unique dualizing module, but it is not isomorphic to "R".
The ring Z[] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
The local ring "k"["x","y"]/("y"2,"xy") is not Cohen–Macaulay so does not have a dualizing module.


Discrete transform

In signal processing, discrete transforms are mathematical transforms, often linear transforms, of signals between discrete domains, such as between discrete time and discrete frequency.
Many common integral transforms used in signal processing have their discrete counterparts. For example, for the Fourier transform the counterpart is the discrete Fourier transform.
In addition to spectral analysis of signals, discrete transforms play important role in data compression, signal detection, digital filtering and correlation analysis. The discrete cosine transform (DCT) is the most widely used transform coding compression algorithm in digital media, followed by the discrete wavelet transform (DWT).
Transforms between a discrete domain and a continuous domain are not discrete transforms. For example, the discrete-time Fourier transform and the Z-transform, from discrete time to continuous frequency, and the Fourier series, from continuous time to discrete frequency, are outside the class of discrete transforms.
Classical signal processing deals with one-dimensional discrete transforms. Other application areas, such as image processing, computer vision, high definition television, visual telephony, etc. make use of two-dimensional and in general, multidimensional discrete transforms.


Higman group

In mathematics, the Higman group, introduced by , was the first example of an infinite finitely presented group with no non-trivial finite quotients.
The quotient by the maximal proper normal subgroup is a finitely generated infinite simple group. later found some finitely presented infinite groups that are simple if is even and have a simple subgroup of index 2 if is odd, one of which is one of the Thompson groups.
Higman's group is generated by 4 elements with the relations


Motion vector

In video compression, a motion vector is the key element in the motion estimation process. It is used to represent a macroblock in a picture based on the position of this macroblock (or a similar one) in another picture, called the reference picture.
The H.264/MPEG-4 AVC standard defines motion vector as:
motion vector: a two-dimensional vector used for inter prediction that provides an offset from the coordinates in the decoded picture to the coordinates in a reference picture.


Bicyclic semigroup

In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic monoid describing the Dyck language of balanced pairs of parentheses. Thus, it finds common applications in combinatorics, such as describing binary trees and associative algebras.
History.
The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H. Clifford and Gordon Preston claim that one of them, working with David Rees, discovered it independently (without publication) at some point before 1943.
Construction.
There are at least three standard ways of constructing the bicyclic semigroup, and various notations for referring to it. Lyapin called it "P"; Clifford and Preston used formula_1; and most recent papers have tended to use "B". This article will use the modern style throughout.
From a free semigroup.
The bicyclic semigroup is the free monoid on two generators "p" and "q", under the relation "p" "q" = 1. That is, each semigroup element is a string of those two letters, with the proviso that the subsequence ""p" "q" does not appear.
The semigroup operation is concatenation of strings, which is clearly associative.
It can then be shown that all elements of "B" in fact have the form "q""a" "p""b", for some natural numbers "a" and "b". The composition operation simplifies to
From ordered pairs.
The way in which these exponents are constrained suggests that the "p" and "q" structure" can be discarded, leaving only operations on the ""a" and "b"" part.
So "B" is the semigroup of pairs of natural numbers (including zero), with operation
This is sufficient to define "B" so that it is the same object as in the original construction. Just as "p" and "q" generated "B" originally, with the empty string as the monoid identity, this new construction of "B" has generators (1, 0) and (0, 1), with identity (0, 0).
From functions.
It can be shown that "any" semigroup "S" generated by elements "e", "a", and "b" satisfying the statements below is isomorphic to the bicyclic semigroup.
It is not entirely obvious that this should be the case—perhaps the hardest task is understanding that "S" must be infinite. To see this, suppose that "a" (say) does not have infinite order, so "a""k" + "h" = "a""h" for some "h" and "k". Then "a""k" = "e", and
so
which is not allowed—so there are infinitely many distinct powers of "a". The full proof is given in Clifford and Preston's book.
Note that the two definitions given above both satisfy these properties. A third way of deriving "B" uses two appropriately-chosen functions to yield the bicyclic semigroup as a monoid of transformations of the natural numbers. Let α, β, and ι be elements of the transformation semigroup on the natural numbers, where
These three functions have the required properties, so the semigroup they generate is "B".
Properties.
The bicyclic semigroup has the property that the image of any homomorphism φ from "B" to another semigroup "S" is either cyclic, or it is an isomorphic copy of "B". The elements φ("a"), φ("b") and φ("e") of "S" will always satisfy the conditions above (because φ is a homomorphism) with the possible exception that φ("b") φ("a") might turn out to be φ("e"). If this is not true, then φ("B") is isomorphic to "B"; otherwise, it is the cyclic semigroup generated by φ("a"). In practice, this means that the bicyclic semigroup can be found in many different contexts.
The idempotents of "B" are all pairs ("x", "x"), where "x" is any natural number (using the ordered pair characterisation of "B"). Since these commute, and "B" is "regular" (for every "x" there is a "y" such that "x" "y" "x" = "x"), the bicyclic semigroup is an inverse semigroup. (This means that each element "x" of "B" has a unique inverse "y", in the "weak" semigroup sense that "x" "y" "x" = "x" and "y" "x" "y" = "y".)
Every ideal of "B" is principal: the left and right principal ideals of ("m", "n") are
Each of these contains infinitely many others, so "B" does not have minimal left or right ideals.
In terms of Green's relations, "B" has only one "D"-class (it is "bisimple"), and hence has only one "J"-class (it is "simple"). The "L" and "R" relations are given by
This implies that two elements are "H"-related if and only if they are identical. Consequently, the only subgroups of "B" are infinitely many copies of the trivial group, each corresponding to one of the idempotents.
The egg-box diagram for "B" is infinitely large; the upper left corner begins:
Each entry represents a singleton "H"-class; the rows are the "R"-classes and the columns are "L"-classes. The idempotents of "B" appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each "L"-class and each "R"-class must contain exactly one idempotent.
The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. Where the definition of "B" from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative lattice under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.


Divisibility (ring theory)

In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension.
Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.
Definition.
Let "R" be a ring, and let "a" and "b" be elements of "R". If there exists an element "x" in "R" with , one says that "a" is a left divisor of "b" and that "b" is a right multiple of "a". Similarly, if there exists an element "y" in "R" with , one says that "a" is a right divisor of "b" and that "b" is a left multiple of "a". One says that "a" is a two-sided divisor of "b" if it is both a left divisor and a right divisor of "b"; the "x" and "y" above are not required to be equal.
When "R" is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that "a" is a divisor of "b", or that "b" is a multiple of "a", and one writes formula_1. Elements "a" and "b" of an integral domain are associates if both formula_1 and formula_3. The associate relationship is an equivalence relation on "R", so it divides "R" into disjoint equivalence classes.
Note: Although these definitions make sense in any magma, they are used primarily when this magma is the multiplicative monoid of a ring.
Properties.
Statements about divisibility in a commutative ring formula_4 can be translated into statements about principal ideals. For instance,
In the above, formula_10 denotes the principal ideal of formula_4 generated by the element formula_12.


Non-negative least squares

In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix and a (column) vector of response variables , the goal is to find
Here means that each component of the vector should be non-negative, and denotes the Euclidean norm.
Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC and non-negative matrix/tensor factorization. The latter can be considered a generalization of NNLS.
Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds .
Quadratic programming version.
The NNLS problem is equivalent to a quadratic programming problem
where = and = . This problem is convex, as is positive semidefinite and the non-negativity constraints form a convex feasible set.
Algorithms.
The first widely used algorithm for solving this problem is an active-set method published by Lawson and Hanson in their 1974 book "Solving Least Squares Problems". In pseudocode, this algorithm looks as follows:
This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse . Variants of this algorithm are available in MATLAB as the routine and in SciPy as .
Many improved algorithms have been suggested since 1974. Fast NNLS (FNNLS) is an optimized version of the Lawson—Hanson algorithm. Other algorithms include variants of Landweber's gradient descent method and coordinate-wise optimization based on the quadratic programming problem above.


Homogeneous linear equation


Nonnegative rank (linear algebra)

In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative.
For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative.
Formal definition.
There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative "m×n"-matrix "A" is equal to the smallest number "q" such there exists a nonnegative "m×q"-matrix "B" and a nonnegative "q×n"-matrix "C" such that "A = BC" (the usual matrix product). To obtain the linear rank, drop the condition that "B" and "C" must be nonnegative.
Further, the nonnegative rank is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively:
where "Rj ≥ 0" stands for ""Rj" is nonnegative". (To obtain the usual linear rank, drop the condition that the "Rj" have to be nonnegative.)
Given a nonnegative formula_1 matrix "A" the nonnegative rank formula_2 of "A" satisfies
A Fallacy.
The rank of the matrix "A" is the largest number of columns which are linearly independent, i.e., none of the selected columns can be written as a linear combination of the other selected columns. It is not true that adding nonnegativity to this characterization gives the nonnegative rank: The nonnegative rank is in general less than or equal to the largest number of columns such that no selected column can be written as a nonnegative linear combination of the other selected columns.
Connection with the linear rank.
It is always true that "rank(A) ≤ rank+(A)". In fact "rank+(A) = rank(A)" holds whenever "rank(A) ≤ 2" [2].
In the case "rank(A) ≥ 3", however, "rank(A) < rank+(A)" is possible. For example, the matrix
formula_3
satisfies "rank(A) = 3 < 4 = rank+(A)" [2].
Computing the nonnegative rank.
The nonnegative rank of a matrix can be determined algorithmically.
It has been proved that determining whether <math>


Hermite ring

In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects.
According to (p. 465), a ring is right Hermite if, for every two elements "a" and "b" of the ring, there is an element "d" of the ring and an invertible 2 by 2 matrix "M" over the ring such that "(a b)M=(d 0)". (The term left Hermite is defined similarly.) Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (, p. 468.) (appendix to §I.4) calls this property K-Hermite, using "Hermite" instead in the sense given below.
According to (§I.4, p. 26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector "(b1...,bn)" of elements of the ring which generate it as a right module (i.e., "b1R+...+bnR=R") can be completed to a (not necessarily square) invertible matrix by adding some number of rows. (The criterion of being left Hermite can be defined similarly.) (p. 528) earlier called a commutative ring with this property an H-ring.
According to (§0.4), a ring is Hermite if, in addition to every stably free (left) module being free, it has IBN.
All commutative rings which are Hermite in the sense of Kaplansky are also Hermite in the sense of Lam, but the converse is not necessarily true. All Bézout domains are Hermite in the sense of Kaplansky, and a commutative ring which is Hermite in the sense of Kaplansky is also a Bézout ring (, pp. 39-40.)
The Hermite ring conjecture, introduced by (p. xi), states that if "R" is a commutative Hermite ring, then "R"["x"] is a Hermite ring.


Special linear group

In mathematics, the special linear group of degree "n" over a field "F" is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
where we write "F"× for the multiplicative group of "F" (that is, "F" excluding 0).
These elements are "special" in that they form a algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
Geometric interpretation.
The special linear group can be characterized as the group of "volume and orientation preserving" linear transformations of R"n"; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.
Lie subgroup.
When "F" is R or C, is a Lie subgroup of of dimension . The Lie algebra formula_2 of SL("n", "F") consists of all matrices over "F" with vanishing trace. The Lie bracket is given by the commutator.
Topology.
Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues. The determinant of the unitary matrix is on the unit circle while that of the hermitian matrix is real and positive and since in the case of a matrix from the special linear group the product of these two determinants must be 1, then each of them must be 1. Therefore, a special linear matrix can be written as the product of a special unitary matrix (or special orthogonal matrix in the real case) and a positive definite hermitian matrix (or symmetric matrix in the real case) having determinant 1.
Thus the topology of the group is the product of the topology of SU("n") and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues. A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of -dimensional Euclidean space. Since SU("n") is simply connected, we conclude that is also simply connected, for all "n".
The topology of is the product of the topology of SO("n") and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of -dimensional Euclidean space. Thus, the group has the same fundamental group as SO("n"), that is, Z for and Z2 for . In particular this means that , unlike , is not simply connected, for "n" greater than 1.
Relations to other subgroups of GL("n","A").
Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [GL, GL] ≤ SL), but in general do not coincide with it.
The group generated by transvections is denoted (for elementary matrices) or . By the second Steinberg relation, for , transvections are commutators, so for , .
For , transvections need not be commutators (of matrices), as seen for example when "A" is F2, the field of two elements, then
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
However, if "A" is a field with more than 2 elements, then , and if "A" is a field with more than 3 elements, .
In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections, and the "stable" special linear group over a Dedekind domain is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group , where SL("A") and E("A") are the stable groups of the special linear group and elementary matrices.
Generators and relations.
If working over a ring where SL is generated by transvections (such as a field or Euclidean domain), one can give a presentation of SL using transvections with some relations. Transvections satisfy the Steinberg relations, but these are not sufficient: the resulting group is the Steinberg group, which is not the special linear group, but rather the universal central extension of the commutator subgroup of GL.
A sufficient set of relations for for is given by two of the Steinberg relations, plus a third relation .
Let (1) be the elementary matrix with 1's on the diagonal and in the "ij" position, and 0's elsewhere (and "i" ≠ "j"). Then
are a complete set of relations for SL("n", Z), "n" ≥ 3.
SL±("n","F").
In characteristic other than 2, the set of matrices with determinant form another subgroup of GL, with SL as an index 2 subgroup (necessarily normal); in characteristic 2 this is the same as SL. This forms a short exact sequence of groups:
This sequence splits by taking any matrix with determinant , for example the diagonal matrix formula_6 If formula_7 is odd, the negative identity matrix formula_8 is in but not in and thus the group splits as an internal direct product formula_9. However, if formula_10 is even, formula_8 is already in , does not split, and in general is a non-trivial group extension.
Over the real numbers, has two connected components, corresponding to and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant ). In odd dimension these are naturally identified by formula_8, but in even dimension there is no one natural identification.
Structure of GL("n","F").
The group splits over its determinant (we use as the monomorphism from "F"× to , see semidirect product), and therefore can be written as a semidirect product of by "F"×:


Quantum q-Krawtchouk polynomials

In mathematics, the quantum "q"-Krawtchouk 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 by


Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers.
Every such quadratic field is some where is a (uniquely defined) square-free integer different form and . If , the corresponding quadratic field is called a real quadratic field, and for an imaginary quadratic field or complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.
Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.
Discriminant.
For a nonzero square free integer "d", the discriminant of the quadratic field "K"=Q() is "d" if "d" is congruent to 1 modulo 4, and otherwise 4"d". For example, if "d" is −1, then "K" is the field of Gaussian rationals and the discriminant is −4. The reason for such a distinction is that the ring of integers of "K" is generated by (1+) in the first case, but by in the second case.
The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.
Prime factorization into ideals.
Any prime number "p" gives rise to an ideal "pO""K" in the ring of integers "O""K" of a quadratic field "K". In line with general theory of splitting of prime ideals in Galois extensions, this may be
The third case happens if and only if "p" divides the discriminant "D". The first and second cases occur when the Kronecker symbol ("D/p") equals −1 and +1, respectively. For example, if "p" is an odd prime not dividing "D", then "p" splits if and only if "D" is congruent to a square modulo "p". The first two cases are in a certain sense equally likely to occur as "p" runs through the primes, see Chebotarev density theorem.
The law of quadratic reciprocity implies that the splitting behaviour of a prime "p" in a quadratic field depends only on "p" modulo "D", where "D" is the field discriminant.
Class group.
Determining the class group of a quadratic field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group. A quadratic field formula_1 has discriminantformula_2so the Minkowski bound isformula_3
Then, the ideal class group is generated by the prime ideals whose norm is less than formula_4. This can be done by looking at the decomposition of the ideals formula_5 for formula_6 prime where formula_7 page 72. These decompositions can be found using the Kummer-Dedekind theorem.
Quadratic subfields of cyclotomic fields.
The quadratic subfield of the prime cyclotomic field.
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive "p"-th root of unity, with "p" a prime number > 2. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index 2 in the Galois group over Q. As explained at Gaussian period, the discriminant of the quadratic field is "p" for "p" = 4"n" + 1 and −"p" for "p" = 4"n" + 3. This can also be predicted from enough ramification theory. In fact "p" is the only prime that ramifies in the cyclotomic field, so that "p" is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants −4"p" and 4"p" in the respective cases.
Other cyclotomic fields.
If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields. In general a quadratic field of field discriminant "D" can be obtained as a subfield of a cyclotomic field of "D"-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
Orders of quadratic number fields of small discriminant.
The following table shows some orders of small discriminant of quadratic fields. The "maximal order" of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the square of the determinant of the matrix that expresses a basis of the non-maximal order over a basis of the maximal order. All these discriminants may be defined by the formula of .
For real quadratic integer rings, the ideal class number, which measures the failure of unique factorization, is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.
Some of these examples are listed in Artin, "Algebra" (2nd ed.), §13.8.


Multilinear map

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
where formula_2 and formula_3 are vector spaces (or modules over a commutative ring), with the following property: for each formula_4, if all of the variables but formula_5 are held constant, then formula_6 is a linear function of formula_5.
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of "k" variables is called a "k"-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating "k"-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
Coordinate representation.
Let
be a multilinear map between finite-dimensional vector spaces, where formula_16 has dimension formula_17, and formula_18 has dimension formula_19. If we choose a basis formula_20 for each formula_16 and a basis formula_22 for formula_18 (using bold for vectors), then we can define a collection of scalars formula_24 by
Then the scalars formula_26 completely determine the multilinear function formula_27. In particular, if
for formula_29, then
Example.
Let's take a trilinear function
where , and .
A basis for each is formula_32 Let
where formula_34. In other words, the constant formula_35 is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three formula_36), namely:
Each vector formula_38 can be expressed as a linear combination of the basis vectors
The function value at an arbitrary collection of three vectors formula_40 can be expressed as
Or, in expanded form as
Relation to tensor products.
There is a natural one-to-one correspondence between multilinear maps
and linear maps
where formula_45 denotes the tensor product of formula_2. The relation between the functions formula_27 and formula_11 is given by the formula
Multilinear functions on "n"×"n" matrices.
One can consider multilinear functions, on an matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as
satisfying
If we let formula_52 represent the th row of the identity matrix, we can express each row as the sum
Using the multilinearity of we rewrite as
Continuing this substitution for each we get, for ,
where, since in our case ,
is a series of nested summations.
Therefore, is uniquely determined by how operates on formula_57.
Example.
In the case of 2×2 matrices we get
Where formula_59 and formula_60. If we restrict formula_61 to be an alternating function then formula_62 and formula_63. Letting formula_64 we get the determinant function on 2×2 matrices: