In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because they are used to classify certain classes of simple semigroups.
Definition.
Let "S" be a semigroup, "I" and "Λ" non-empty sets and "P" a matrix indexed by "I" and "Λ" with entries "p""i","λ" taken from "S".
Then the Rees matrix semigroup "M"("S"; "I", "Λ"; "P") is the set "I"×"S"×"Λ" together with the multiplication
Rees matrix semigroups are an important technique for building new semigroups out of old ones.
Rees' theorem.
In his 1940 paper Rees proved the following theorem characterising completely simple semigroups:
That is, every completely simple semigroup is isomorphic to a semigroup of the form "M"("G"; "I", "Λ"; "P") for some group "G". Moreover, Rees proved that if "G" is a group and "G"0 is the semigroup obtained from "G" by attaching a zero element, then "M"("G"0; "I", "Λ"; "P") is a regular semigroup if and only if every row and column of the matrix "P" contains an element which is not 0. If such an "M"("G"0; "I", "Λ"; "P") is regular, then it is also completely 0-simple.


Loewy ring

In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy.
Loewy length.
The Loewy length and Loewy series were introduced by
If "M" is a module, then define the Loewy series "M"α for ordinals α by "M"0 = 0, "M"α+1/"M"α = socle "M"/"M"α, "M"α = ∪λ<α "M"λ if α is a limit ordinal. The Loewy length of "M" is defined to be the smallest α with "M" = "M"α, if it exists.
Semiartinian modules.
formula_1 is a semiartinian module if, for all formula_2 epimorphism, where formula_3, the socle of formula_4 is essential in formula_4.
Note that if formula_1 is an artinian module then formula_1 is a semiartinian module. Clearly 0 is semiartinian.
Let formula_8 be exact then formula_9 and formula_10 are semiartinian if and only if formula_11 is semiartinian.
Consider formula_12 family of formula_13-modules, then formula_14 is semiartinian if and only if formula_15 is semiartinian for all formula_16.
Semiartinian rings.
formula_13 is called left semiartinian if formula_18 is semiartinian, that is, formula_13 is left semiartinian if for any left ideal formula_20, formula_21 contains a simple submodule.
Note that formula_13 left semiartinian does not imply formula_13 left artinian.


Characteristic 2 type

In mathematical finite group theory, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2.
In the classification of finite simple groups, there is a major division between group of characteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements.
Groups of characteristic 2 type and rank at least 3 are classified by the trichotomy theorem.
Definitions.
A group is said to be of even characteristic if
If this condition holds for all maximal 2-local subgroups "M" then "G" is said to be of characteristic 2 type.
 use a modified version of this called even type.


Absolute Galois group

In mathematics, the absolute Galois group "GK" of a field "K" is the Galois group of "K"sep over "K", where "K"sep is a separable closure of "K". Alternatively it is the group of all automorphisms of the algebraic closure of "K" that fix "K". The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.


Sheldon Katz

Sheldon H. Katz (19 December 1956, Brooklyn) is an American mathematician, specializing in algebraic geometry and its applications to string theory.
Background and Career.
In 1973 Katz won first prize in the U.S.A. Mathematical Olympiad. He received in 1976 his bachelor's degree from MIT and in 1980 his Ph.D. from Princeton University under Robert C. Gunning with thesis "Deformations of Linear Systems, Divisors and Weierstrass Points on Curves". At the University of Utah, he was an instructor from 1980 to 1984. At the University of Oklahoma he was an assistant professor from 1984 to 1987. At Oklahoma State University, he became in 1987 an assistant professor, in 1989 an associate professor, in 1994 a full professor, in 1997 "Southwestern Bell Professor", and in 1999 "Regents Professor." Since 2001 he has been a professor at the University of Illinois, Urbana-Champaign, where he was chair of the department in 2006–2011.
For the academic year 1982/83 he was a visiting scholar at the Institute for Advanced Study. He was a visiting professor at the Mittag-Leffler Institute (1997), at Duke University (1991/92) and at the University of Bayreuth (1989).
His research on algebraic geometry and its applications to string theory (including mirror symmetry) and supersymmetry has been published in prestigious journals in mathematics and physics.
In 2013 he was elected a Fellow of the American Mathematical Society.


Commensurability (group theory)

In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
Commensurability in group theory.
Two groups "G"1 and "G"2 are said to be (abstractly) commensurable if there are subgroups "H"1 ⊂ "G"1 and "H"2 ⊂ "G"2 of finite index such that "H"1 is isomorphic to "H"2. For example:
A different but related notion is used for subgroups of a given group. Namely, two subgroups Γ1 and Γ2 of a group "G" are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. Clearly this implies that Γ1 and Γ2 are abstractly commensurable.
Example: for nonzero real numbers "a" and "b", the subgroup of R generated by "a" is commensurable with the subgroup generated by "b" if and only if the real numbers "a" and "b" are commensurable, meaning that "a"/"b" belongs to the rational numbers Q.
In geometric group theory, a finitely generated group is viewed as a metric space using the word metric. If two groups are (abstractly) commensurable, then they are quasi-isometric. It has been fruitful to ask when the converse holds.
There is an analogous notion in linear algebra: two linear subspaces "S" and "T" of a vector space "V" are commensurable if the intersection "S" ∩ "T" has finite codimension in both "S" and "T".
In topology.
Two path-connected topological spaces are sometimes called "commensurable" if they have homeomorphic finite-sheeted covering spaces. Depending on the type of space under consideration, one might want to use homotopy equivalences or diffeomorphisms instead of homeomorphisms in the definition. By the relation between covering spaces and the fundamental group, commensurable spaces have commensurable fundamental groups.
Example: the Gieseking manifold is commensurable with the complement of the figure-eight knot; these are both noncompact hyperbolic 3-manifolds of finite volume. On the other hand, there are infinitely many different commensurability classes of compact hyperbolic 3-manifolds, and also of noncompact hyperbolic 3-manifolds of finite volume.
The commensurator.
The commensurator of a subgroup Γ of a group "G", denoted Comm"G"(Γ), is the set of elements "g" of "G" that such that the conjugate subgroup "g"Γ"g"−1 is commensurable with Γ. In other words,
This is a subgroup of "G" that contains the normalizer N"G"(Γ) (and hence contains Γ).
For example, the commensurator of the special linear group "SL"("n",Z) in "SL"("n",R) contains "SL"("n",Q). In particular, the commensurator of "SL"("n",Z) in "SL"("n",R) is dense in "SL"("n",R). More generally, Grigory Margulis showed that the commensurator of a lattice Γ in a semisimple Lie group "G" is dense in "G" if and only if Γ is an arithmetic subgroup of "G".
The abstract commensurator.
The abstract commensurator of a group formula_2, denoted Commformula_3, is the group of equivalence classes of isomorphisms formula_4, where formula_5 and formula_6 are finite index subgroups of formula_2, under composition. Elements of formula_8 are called commensurators of formula_2.
If formula_2 is a connected semisimple Lie group not isomorphic to formula_11, with trivial center and no compact factors, then by the Mostow rigidity theorem, the abstract commensurator of any irreducible lattice formula_12 is linear. Moreover, if formula_13 is arithmetic, then Commformula_14 is virtually isomorphic to a dense subgroup of formula_2, otherwise Commformula_14 is virtually isomorphic to formula_13.


Gassmann triple

In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a group "G" together with two faithful actions on sets "X" and "Y", such that "X" and "Y" are not isomorphic as "G"-sets but every element of "G" has the same number of fixed points on "X" and "Y". They were introduced by Fritz Gassmann in 1926.
Applications.
Gassmann triples have been used to construct examples of pairs of mathematical objects with the same invariants that are not isomorphic, including arithmetically equivalent number fields and isospectral graphs and isospectral Riemannian manifolds.
Examples.
The simple group "G"  =  SL3(F2) of order 168 acts on the projective plane of order 2, and the actions on the 7 points and 7 lines give a Gassmann triple.


General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the "general case" situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.
For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case.
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see "generic complexity").
General linear position.
A set of points in a -dimensional affine space (-dimensional Euclidean space is a common example) is in general linear position (or just general position) if no of them lie in a -dimensional flat for . These conditions contain considerable redundancy since, if the condition holds for some value then it also must hold for all with . Thus, for a set containing at least points in -dimensional affine space to be in general position, it suffices to know that no hyperplane contains more than points — "i.e." the points do not satisfy any more linear relations than they must.
A set of at most points in general linear position is also said to be "affinely independent" (this is the affine analog of linear independence of vectors, or more precisely of maximal rank), and points in general linear position in affine "d"-space are an affine basis. See affine transformation for more.
Similarly, "n" vectors in an "n"-dimensional vector space are linearly independent if and only if the points they define in projective space (of dimension ) are in general linear position.
If a set of points is not in general linear position, it is called a degenerate case or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.
A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).
More generally.
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections). In algebraic geometry this kind of condition is frequently encountered, in that points should impose "independent" conditions on curves passing through them.
For example, five points determine a conic, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.
General position is preserved under biregular maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the Veronese map is biregular; as points under the Veronese map corresponds to evaluating a degree "d" polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.
The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for higher degree.
This is not sufficient, however. While nine points determine a cubic, there are configurations of nine points that are special with respect to cubics, namely the intersection of two cubics. The intersection of two cubics, which is formula_1 points (by Bézout's theorem), is special in that nine points in general position are contained in a "unique" cubic, while if they are contained in two cubics they in fact are contained in a pencil (1-parameter linear system) of cubics, whose equations are the projective linear combinations of the equations for the two cubics. Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the Cayley–Bacharach theorem that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.
For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces.
Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any "k" distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.
Different geometries.
Different geometries allow different notions of geometric constraints. For example, a circle is a concept that makes sense in Euclidean geometry, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse. Similarly, a parabola is a concept in affine geometry but not in projective geometry, where a parabola is simply a kind of conic. The geometry that is overwhelmingly used in algebraic geometry is projective geometry, with affine geometry finding significant but far less use.
Thus, in Euclidean geometry three non-collinear points determine a circle (as the circumcircle of the triangle they define), but four points in general do not (they do so only for cyclic quadrilaterals), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by contrast, circles are not distinct from conics, and five points determine a conic, so there is no projective notion of "general position with respect to circles".
General type.
General position is a property of configurations of points, or more generally other subvarieties (lines in general position, so no three concurrent, and the like). General position is an "extrinsic" notion, which depends on an embedding as a subvariety. Informally, subvarieties are in general position if they cannot be described more simply than others. An "intrinsic" analog of general position is general type, and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of Kodaira dimension of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces (formula_2), and similar classifications occur in higher dimensions, notably the Enriques–Kodaira classification of algebraic surfaces.
Other contexts.
In intersection theory, both in algebraic geometry and in geometric topology, the analogous notion of transversality is used: subvarieties in general intersect "transversally," meaning with multiplicity 1, rather than being tangent or other, higher order intersections.
General position for Delaunay triangulations in the plane.
When discussing Voronoi tessellations and Delaunay triangulations in the plane, a set of points in the plane is said to be in general position only if no four of them lie on the same circle and no three of them are collinear. The usual lifting transform that relates the Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point "p" an extra coordinate equal to |"p"|2) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are "not" in general linear position.
Abstractly: configuration spaces.
In very abstract terms, general position is a discussion of generic properties of a configuration space; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set.
This notion coincides with the measure theoretic notion of generic, meaning almost everywhere on the configuration space, or equivalently that points chosen at random will almost surely (with probability 1) be in general position.


Klein four-group

In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one.
It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation),
as the group of bitwise exclusive or operations on two-bit binary values,
or more abstractly as , the direct product of two copies of the cyclic group of order 2.
It was named Vierergruppe (meaning four-group) by Felix Klein in 1884.
It is also called the Klein group, and is often symbolized by the letter V or as K4.
The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups. The smallest non-abelian group is the symmetric group of degree 3, which has order 6.
Presentations.
The Klein group's Cayley table is given by:
The Klein four-group is also defined by the group presentation
All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, i.e. D4 (or D2, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.
The Klein four-group is also isomorphic to the direct sum , so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.e. over a field of sets with four elements, e.g. formula_2; the empty set is the group's identity element in this case.
Another numerical construction of the Klein four-group is the set with the operation being multiplication modulo 8. Here "a" is 3, "b" is 5, and is .
The Klein four-group has a representation as 2×2 real matrices with the operation being matrix multiplication:
Geometry.
Geometrically, in two dimensions the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.
In three dimensions there are three different symmetry groups that are algebraically the Klein four-group V:
Permutation representation.
The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements.
The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:
In this representation, V is a normal subgroup of the alternating group A4
(and also the symmetric group S4) on four letters. In fact, it is the kernel of a surjective group homomorphism from S4 to S3.
Other representations within S4 are:
They are not normal subgroups of S4.
Algebra.
According to Galois theory, the existence of the Klein four-group (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari:
the map corresponds to the resolvent cubic, in terms of Lagrange resolvents.
In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.
If R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring , and is a subgroup of (in fact it is the component of the identity of ). The quotient group is isomorphic to the Klein four-group. In a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group.
Graph theory.
The simplest simple connected graph that admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities. These include the graph with four vertices and one edge, which remains simple but loses connectivity, and the graph with two vertices connected to each other by two edges, which remains connected but loses simplicity.
Music.
In music composition the four-group is the basic group of permutations in the twelve-tone technique. In that instance the Cayley table is written;


Jan Saxl

Jan Saxl (5 June 1948 – 2 May 2020) was a Czech-British mathematician, and a professor at the University of Cambridge. He was known for his work in finite group theory, particularly on consequences of the classification of finite simple groups.
Education and career.
Saxl was born in Brno, in what was at the time Czechoslovakia. He came to the United Kingdom in 1968, during the Prague Spring.
After undergraduate studies at the University of Bristol, he completed his DPhil in 1973 at the University of Oxford under the direction of Peter M. Neumann, with the title of "Multiply Transitive Permutation Groups".
Saxl held postdoctoral positions at Oxford and the University of Illinois at Chicago, and a lecturer position at the University of Glasgow. He moved to the University of Cambridge in 1976, and spent the rest of his career there. He was elected as a fellow of Gonville and Caius College in 1986, and he retired in 2015.
Saxl published around 100 papers, and according to MathSciNet, these have been cited over 1900 times. He is noted for his work in finite group theory, particularly on permutation groups, and often coauthored with Robert Guralnick, Martin Liebeck, and Cheryl Praeger. Some notable and highly-cited examples of this work are as follows. Liebeck, Saxl and Praeger gave a relatively simple and self-contained proof of the O'Nan–Scott theorem. It had long been known that every maximal subgroup of a symmetric group or alternating group was intransitive, imprimitive, or primitive, and the same authors in 1988 gave a partial description of which primitive subgroups could occur.
Personal life.
Saxl was married to Cambridge mathematician Ruth M. Williams and they had one daughter, Miriam.
Death.
Saxl died on 2 May 2020, after a long period of poor health.
Awards and honors.
A three-day conference in the joint honor of Saxl and Martin Liebeck was held at the University of Cambridge in July 2015.
Publications.
Books
Selected articles


Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.
Definition.
Let "C" be a category. A "congruence relation" "R" on "C" is given by: for each pair of objects "X", "Y" in "C", an equivalence relation "R""X","Y" on Hom("X","Y"), such that the equivalence relations respect composition of morphisms. That is, if
are related in Hom("X", "Y") and
are related in Hom("Y", "Z"), then "g"1"f"1 and "g"2"f"2 are related in Hom("X", "Z").
Given a congruence relation "R" on "C" we can define the quotient category "C"/"R" as the category whose objects are those of "C" and whose morphisms are equivalence classes of morphisms in "C". That is,
Composition of morphisms in "C"/"R" is well-defined since "R" is a congruence relation.
Properties.
There is a natural quotient functor from "C" to "C"/"R" which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).
Every functor "F" : "C" → "D" determines a congruence on "C" by saying "f" ~ "g" iff "F"("f") = "F"("g"). The functor "F" then factors through the quotient functor "C" → "C"/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for functors.
Related concepts.
Quotients of additive categories modulo ideals.
If "C" is an additive category and we require the congruence relation ~ on "C" to be additive (i.e. if "f"1, "f"2, "g"1 and "g"2 are morphisms from "X" to "Y" with "f"1 ~ "f"2 and "g"1 ~"g"2, then "f"1 + "f"2 ~ "g"1 + "g"2), then the quotient category "C"/~ will also be additive, and the quotient functor "C" → "C"/~ will be an additive functor.
The concept of an additive congruence relation is equivalent to the concept of a "two-sided ideal of morphisms": for any two objects "X" and "Y" we are given an additive subgroup "I"("X","Y") of Hom"C"("X", "Y") such that for all "f" ∈ "I"("X","Y"), "g" ∈ Hom"C"("Y", "Z") and "h"∈ Hom"C"("W", "X"), we have "gf" ∈ "I"("X","Z") and "fh" ∈ "I"("W","Y"). Two morphisms in Hom"C"("X", "Y") are congruent iff their difference is in "I"("X","Y").
Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.
Localization of a category.
The localization of a category introduces new morphisms to turn several of the original category's morphisms into isomorphisms. This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories. But in both constructions it often happens that two objects become isomorphic that weren't isomorphic in the original category.
Serre quotients of abelian categories.
The Serre quotient of an abelian category by a Serre subcategory is a new abelian category which is similar to a quotient category but also in many cases has the character of a localization of the category.


Compound matrix

In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras.
Definition.
Let be an matrix with real or complex entries. If is a subset of and is a subset of , then the -submatrix of , written , is the submatrix formed from by retaining only those rows indexed by and those columns indexed by . If , then is the -minor of .
The "r"th compound matrix of is a matrix, denoted , is defined as follows. If , then is the unique matrix. Otherwise, has size formula_1. Its rows and columns are indexed by -element subsets of and , respectively, in their lexicographic order. The entry corresponding to subsets and is the minor .
In some applications of compound matrices, the precise ordering of the rows and columns is unimportant. For this reason, some authors do not specify how the rows and columns are to be ordered.
For example, consider the matrix
The rows are indexed by and the columns by . Therefore, the rows of are indexed by the sets
and the columns are indexed by
Using absolute value bars to denote determinants, the second compound matrix is
Properties.
Let be a scalar, be an matrix, and be an matrix. If is a positive integer, then denotes the identity matrix. The transpose of a matrix will be written , and the conjugate transpose by . Then:
Assume in addition that is a square matrix of size . Then:
Relation to exterior powers.
Give the standard coordinate basis . The th exterior power of is the vector space
whose basis consists of the formal symbols
where
Suppose that be an matrix. Then corresponds to a linear transformation
Taking the th exterior power of this linear transformation determines a linear transformation
The matrix corresponding to this linear transformation (with respect to the above bases of the exterior powers) is . Taking exterior powers is a functor, which means that
This corresponds to the formula . It is closely related to, and is a strengthening of, the Cauchy–Binet formula.
Relation to adjugate matrices.
Let be an matrix. Recall that its th higher adjugate matrix is the formula_14 matrix whose entry is
where, for any set of integers, is the sum of the elements of . The adjugate of is its 1st higher adjugate and is denoted . The generalized Laplace expansion formula implies
If is invertible, then
A concrete consequence of this is Jacobi's formula for the minors of an inverse matrix:
Adjugates can also be expressed in terms of compounds. Let denote the "sign matrix":
and let denote the "exchange matrix":
Then Jacobi's theorem states that the th higher adjugate matrix is:
It follows immediately from Jacobi's theorem that
Taking adjugates and compounds does not commute. However, compounds of adjugates can be expressed using adjugates of compounds, and vice versa. From the identities
and the Sylvester-Franke theorem, we deduce
The same technique leads to an additional identity,
Applications.
The computation of compound matrices appears in a wide array of problems.
Compound and adjugate matrices appear when computing determinants of linear combinations of matrices. It is elementary to check that, if and are matrices, then
It is also true that:
This has the immediate consequence
Numerical computation.
In general, the computation of compound matrices is non effective due to its high complexity. Nonetheless, there is some efficient algorithms available for real matrices with special structures.


Signalizer functor

In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a formula_1-subgroup of a finite group formula_2, which has a good chance of being normal in formula_2, by taking as generators certain formula_1-subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian formula_5-subgroups of formula_6 The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including who defined signalizer functors, who proved the Solvable Signalizer Functor Theorem for solvable groups, and who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups.
Definition.
Let "A" be a noncyclic elementary abelian "p"-subgroup of the finite group "G." An A-signalizer functor on "G" or simply a signalizer functor when "A" and "G" are clear is a mapping "θ" from the set of nonidentity elements of "A" to the set of "A"-invariant "p′"-subgroups of "G" satisfying the following properties:
The second condition above is called the balance condition. If the subgroups formula_12 are all solvable, then the signalizer functor formula_13 itself is said to be solvable.
Solvable signalizer functor theorem.
Given formula_14 certain additional, relatively mild, assumptions allow one to prove that the subgroup formula_15 of formula_2 generated by the subgroups formula_12 is in fact a formula_1-subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if formula_13 is solvable and formula_20 has at least three generators. The theorem also states that under these assumptions, formula_21 itself will be solvable.
Several earlier versions of the theorem were proven: proved this under the stronger assumption that formula_20 had rank at least 5. proved this under the assumption that formula_20 had rank at least 4 or was a 2-group of rank at least 3. gave a simple proof for 2-groups using the ZJ theorem, and a proof in a similar spirit has been given for all primes by . gave the definitive result for solvable signalizer functors. Using the classification of finite simple groups, showed that formula_21 is a formula_1-group without the assumption that formula_13 is solvable.
Completeness.
The terminology of completeness is often used in discussions of signalizer functors. Let formula_13 be a signalizer functor as above, and consider the set И of all formula_20-invariant formula_1-subgroups formula_30 of formula_2 satisfying the following condition:
For example, the subgroups formula_12 belong to И by the balance condition. The signalizer functor formula_13 is said to be complete if И has a unique maximal element when ordered by containment. In this case, the unique maximal element can be shown to coincide with formula_21 above, and formula_21 is called the completion of formula_13. If formula_13 is complete, and formula_21 turns out to be solvable, then formula_13 is said to be solvably complete.
Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if formula_20 has at least three generators, then every solvable formula_20-signalizer functor on formula_2 is solvably complete.
Examples of signalizer functors.
The easiest way to obtain a signalizer functor is to start with an formula_20-invariant formula_1-subgroup formula_47 of formula_48 and define formula_49 for all nonidentity formula_33 In practice, however, one begins with formula_13 and uses it to construct the formula_20-invariant formula_1-group.
The simplest signalizer functor used in practice is this:
formula_54
A few words of caution are needed here. First, note that formula_12 as defined above is indeed an formula_20-invariant formula_1-subgroup of formula_2 because formula_20 is abelian. However, some additional assumptions are needed to show that this formula_13 satisfies the balance condition. One sufficient criterion is that for each nonidentity formula_61 the group formula_62 is solvable (or formula_5-solvable or even formula_5-constrained). Verifying the balance condition for this formula_13 under this assumption requires a famous lemma, known as Thompson's formula_66-lemma. (Note, this lemma is also called Thompson's formula_67-lemma, but the formula_20 in this use must not be confused with the formula_20 appearing in the definition of a signalizer functor!)
Coprime action.
To obtain a better understanding of signalizer functors, it is essential to know the following general fact about finite groups:
formula_74
To prove this fact, one uses the Schur–Zassenhaus theorem to show that for each prime formula_75 dividing the order of formula_76 the group formula_73 has an formula_70-invariant Sylow formula_75-subgroup. This reduces to the case where formula_73 is a formula_75-group. Then an argument by induction on the order of formula_73 reduces the statement further to the case where formula_73 is elementary abelian with formula_70 acting irreducibly. This forces the group formula_85 to be cyclic, and the result follows. See either of the books or for details.
This is used in both the proof and applications of the Solvable Signalizer Functor Theorem. To begin, notice that it quickly implies the claim that if formula_13 is complete, then its completion is the group formula_21 defined above.
Normal completion.
The completion of a signalizer functor has a "good chance" of being normal in formula_48 according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let formula_13 be a complete formula_20-signalizer functor on formula_2
Let formula_92 be a noncyclic subgroup of formula_93 Then the coprime action fact together with the balance condition imply that
formula_94.
To see this, observe that because formula_12 is "B"-invariant, we have
formula_96
The equality above uses the coprime action fact, and the containment uses the balance condition. Now, it is often the case that formula_13 satisfies an "equivariance" condition, namely that for each formula_98 and nonidentity formula_99
formula_100
The superscript denotes conjugation by formula_101 For example, the mapping formula_102 (which is often a signalizer functor!) satisfies this condition. If formula_13 satisfies equivariance, "then the normalizer of formula_92 will normalize formula_105" It follows that if formula_2 is generated by the normalizers of the noncyclic subgroups of formula_107 then the completion of formula_13 (i.e. W) is normal in formula_6


Complex representation

In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real.
For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation.


Hankel transform

In mathematics, the Hankel transform expresses any given function "f"("r") as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor "k" along the "r" axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor "k" constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.
Definition.
The Hankel transform of order formula_1 of a function "f"("r") is given by
where formula_3 is the Bessel function of the first kind of order formula_1 with formula_5. The inverse Hankel transform of is defined as
which can be readily verified using the orthogonality relationship described below.
Domain of definition.
Inverting a Hankel transform of a function "f"("r") is valid at every point at which "f"("r") is continuous, provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and
However, like the Fourier transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example formula_8.
Alternative definition.
An alternative definition says that the Hankel transform of "g"("r") is
The two definitions are related:
This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse:
The obvious domain now has the condition
but this can be extended. According to the reference given above, we can take the integral as the limit as the upper limit goes to infinity (an improper integral rather than a Lebesgue integral), and in this way the Hankel transform and its inverse work for all functions in L2(0, ∞).
Transforming Laplace's equation.
The Hankel transform can be used to transform and solve Laplace's equation expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by formula_14. In the axisymmetric case, the partial differential equation is transformed as
which is an ordinary differential equation in the transformed variable formula_16.
Orthogonality.
The Bessel functions form an orthogonal basis with respect to the weighting factor "r":
The Plancherel theorem and Parseval's theorem.
If "f"("r") and "g"("r") are such that their Hankel transforms and are well defined, then the Plancherel theorem states
Parseval's theorem, which states
is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property.
Relation to the multidimensional Fourier transform.
The Hankel transform appears when one writes the multidimensional Fourier transform in hyperspherical coordinates, which is the reason why the Hankel transform often appears in physical problems with cylindrical or spherical symmetry.
Consider a function formula_20 of a formula_21-dimensional vector . Its formula_21-dimensional Fourier transform is defined asformula_23To rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into formula_21-dimensional hyperspherical harmonics formula_25:formula_26where formula_27 and formula_28 are the sets of all hyperspherical angles in the formula_29-space and formula_30-space. This gives the following expression for the formula_21-dimensional Fourier transform in hyperspherical coordinates:formula_32If we expand formula_20 and formula_34 in hyperspherical harmonics:formula_35the Fourier transform in hyperspherical coordinates simplifies toformula_36This means that functions with angular dependence in form of a hyperspherical harmonic retain it upon the multidimensional Fourier transform, while the radial part undergoes the Hankel transform (up to some extra factors like formula_37).
Special cases.
Fourier transform in two dimensions.
If a two-dimensional function is expanded in a multipole series,
then its two-dimensional Fourier transform is given byformula_39whereformula_40is the formula_41-th order Hankel transform of formula_42 (in this case formula_41 plays the role of the angular momentum, which was denoted by formula_44 in the previous section).
Fourier transform in three dimensions.
If a three-dimensional function is expanded in a multipole series over spherical harmonics,
then its three-dimensional Fourier transform is given byformula_46whereformula_47is the Hankel transform of formula_48 of order formula_49.
This kind of Hankel transform of half-integer order is also known as the spherical Bessel transform.
Fourier transform in dimensions (radially symmetric case).
If a -dimensional function does not depend on angular coordinates, then its -dimensional Fourier transform also does not depend on angular coordinates and is given byformula_50which is the Hankel transform of formula_51 of order formula_52 up to a factor of formula_53.
2D functions inside a limited radius.
If a two-dimensional function is expanded in a multipole series and the expansion coefficients are sufficiently smooth near the origin and zero outside a radius , the radial part may be expanded into a power series of :
such that the two-dimensional Fourier transform of becomes
where the last equality follows from §6.567.1 of. The expansion coefficients are accessible with discrete Fourier transform techniques: if the radial distance is scaled with
the Fourier-Chebyshev series coefficients emerge as
Using the re-expansion
yields expressed as sums of .
This is one flavor of fast Hankel transform techniques.
Relation to the Fourier and Abel transforms.
The Hankel transform is one member of the FHA cycle of integral operators. In two dimensions, if we define as the Abel transform operator, as the Fourier transform operator, and as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that
In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.
Numerical evaluation.
A simple and efficient approach to the numerical evaluation of the Hankel transform is based on the observation that it can be cast in the form of a convolution by a logarithmic change of variablesformula_60In these new variables, the Hankel transform readsformula_61whereformula_62Now the integral can be calculated numerically with formula_63 complexity using fast Fourier transform. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of formula_64:formula_65The optimal choice of parameters formula_66 depends on the properties of formula_67, in particular its asymptotic behavior at formula_68 and formula_69.
This algorithm is known as the "quasi-fast Hankel transform", or simply "fast Hankel transform".
Since it is based on fast Fourier transform in logarithmic variables, formula_67 has to be defined on a logarithmic grid. For functions defined on a uniform grid, a number of other algorithms exist, including straightforward quadrature, methods based on the projection-slice theorem, and methods using the asymptotic expansion of Bessel functions.
Some Hankel transform pairs.
}{k}</math>
 is a modified Bessel function of the second kind.
 is the complete elliptic integral of the first kind.
The expression
coincides with the expression for the Laplace operator in polar coordinates applied to a spherically symmetric function .
The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976):
for even .


Ideal class group

In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of .
The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.
History and origin of the ideal class group.
Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time.
Later Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's last theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization, i.e., the fundamental theorem of arithmetic, to hold in the rings generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the "p"-torsion in that group for the field of "p"-roots of unity, for any prime number "p", as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime).
Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal domain if and only if it has a trivial ideal class group.
Definition.
If "R" is an integral domain, define a relation ~ on nonzero fractional ideals of "R" by "I" ~ "J" whenever there exist nonzero elements "a" and "b" of "R" such that ("a")"I" = ("b")"J". (Here the notation ("a") means the principal ideal of "R" consisting of all the multiples of "a".) It is easily shown that this is an equivalence relation. The equivalence classes are called the "ideal classes" of "R".
Ideal classes can be multiplied: if ["I"] denotes the equivalence class of the ideal "I", then the multiplication ["I"]["J"] = ["IJ"] is well-defined and commutative. The principal ideals form the ideal class ["R"] which serves as an identity element for this multiplication. Thus a class ["I"] has an inverse ["J"] if and only if there is an ideal "J" such that "IJ" is a principal ideal. In general, such a "J" may not exist and consequently the set of ideal classes of "R" may only be a monoid.
However, if "R" is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of "R". The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except "R") is a product of prime ideals.
Properties.
The ideal class group is trivial (i.e. has only one element) if and only if all ideals of "R" are principal. In this sense, the ideal class group measures how far "R" is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains).
The number of ideal classes (the of "R") may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. But if "R" is in fact a ring of algebraic integers, then the class number is always "finite". This is one of the main results of classical algebraic number theory.
Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski's bound. This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.
The mapping from rings of integers "R" to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with "K"0("R") being the functor assigning to "R" its ideal class group; more precisely, "K"0("R") = Z×"C"("R"), where "C"("R") is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.
Relation with the group of units.
It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. The other part of the answer is provided by the multiplicative group of units of the Dedekind domain, since passage from principal ideals
to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well):
Define a map from "R"× to the set of all nonzero fractional ideals of "R" by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of "R", and its cokernel is the ideal class group of "R". The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.
Examples of ideal class groups.
Class numbers of quadratic fields.
If "d" is a square-free integer (a product of distinct primes) other than 1, then Q() is a quadratic extension of Q. If "d" < 0, then the class number of the ring "R" of algebraic integers of Q() is equal to 1 for precisely the following values of "d": "d" = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. (See Stark-Heegner theorem.) This is a special case of the famous class number problem.
If, on the other hand, "d" > 0, then it is unknown whether there are infinitely many fields Q() with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many number fields with class number 1.
For "d" < 0, the ideal class group of Q() is isomorphic to the class group of integral binary quadratic forms of discriminant equal to the discriminant of Q(). For "d" > 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the narrow class group of Q().
For real quadratic integer rings, the class number is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.
Example of a non-trivial class group.
The quadratic integer ring "R" = Z[] is the ring of integers of Q(). It does "not" possess unique factorization; in fact the class group of "R" is cyclic of order 2. Indeed, the ideal
is not principal, which can be proved by contradiction as follows. formula_1 has a norm function formula_2, which satisfies formula_3, and formula_4 if and only if formula_5 is a unit in formula_1. First of all, formula_7, because the quotient ring of formula_1 modulo the ideal formula_9 is isomorphic to formula_10, so that the quotient ring of formula_1 modulo formula_12 is isomorphic to formula_13. If "J" were generated by an element "x" of "R", then "x" would divide both 2 and 1 + . Then the norm formula_14 would divide both formula_15 and formula_16, so "N"(x) would divide 2. If formula_17, then formula_18 is a unit, and formula_19, a contradiction. But formula_14 cannot be 2 either, because "R" has no elements of norm 2, because the Diophantine equation formula_21 has no solutions in integers, as it has no solutions modulo 5.
One also computes that "J"2 = (2), which is principal, so the class of "J" in the ideal class group has order two. Showing that there aren't any "other" ideal classes requires more effort.
The fact that this "J" is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles:
Connections to class field theory.
Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field "L" of a number field "K" is unique and has the following properties:
Neither property is particularly easy to prove.


Tensor product of fields

In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring, is "never" itself a field. Nonetheless, it is often required to "join" two fields "K" and "L", either in cases where "K" and "L" are given as subfields of a larger field "M", or when "K" and "L" are both field extensions of a smaller field "N" (for example a prime field).
The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents (see radical of a ring).
If "K" and "L" do not have isomorphic prime fields, or in other words they have different characteristics, they have no possibility of being common subfields of a field "M". Correspondingly their tensor product will in that case be the trivial ring (collapse of the construction to nothing of interest).
Compositum of fields.
First, one defines the notion of the compositum of fields. This construction occurs frequently in field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a tower of fields. Let "k" be a field and "L" and "K" be two extensions of "k". The compositum, denoted "K.L" is defined to be formula_1 where the right-hand side denotes the extension generated by "K" and "L". Note that this assumes "some" field containing both "K" and "L". Either one starts in a situation where an ambient field is easy to identify (for example if "K" and "L" are both subfields of the complex numbers), or one proves a result that allows one to place both "K" and "L" (as isomorphic copies) in some large enough field.
In many cases one can identify "K"."L" as a vector space tensor product, taken over the field "N" that is the intersection of "K" and "L". For example, if one adjoins √2 to the rational field ℚ to get "K", and √3 to get "L", it is true that the field "M" obtained as "K"."L" inside the complex numbers ℂ is (up to isomorphism)
as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.)
Subfields "K" and "L" of "M" are linearly disjoint (over a subfield "N") when in this way the natural "N"-linear map of
to "K"."L" is injective. Naturally enough this isn't always the case, for example when "K" = "L". When the degrees are finite, injective is equivalent here to bijective. Hence, when "K" and "L" are linearly disjoint finite-degree extension fields over "N", formula_4, as with the aforementioned extensions of the rationals.
A significant case in the theory of cyclotomic fields is that for the "n"th roots of unity, for "n" a composite number, the subfields generated by the "p""k"th roots of unity for prime powers dividing "n" are linearly disjoint for distinct "p".
The tensor product as ring.
To get a general theory, one needs to consider a ring structure on formula_5. One can define the product formula_6 to be formula_7 (see tensor product of algebras). This formula is multilinear over "N" in each variable; and so defines a ring structure on the tensor product, making formula_5 into a commutative "N"-algebra, called the tensor product of fields.
Analysis of the ring structure.
The structure of the ring can be analysed by considering all ways of embedding both "K" and "L" in some field extension of "N". Note that the construction here assumes the common subfield "N"; but does not assume "a priori" that "K" and "L" are subfields of some field "M" (thus getting round the caveats about constructing a compositum field). Whenever one embeds "K" and "L" in such a field "M", say using embeddings α of "K" and β of "L", there results a ring homomorphism γ from formula_5 into "M" defined by:
The kernel of γ will be a prime ideal of the tensor product; and conversely any prime ideal of the tensor product will give a homomorphism of "N"-algebras to an integral domain (inside a field of fractions) and so provides embeddings of "K" and "L" in some field as extensions of (a copy of) "N".
In this way one can analyse the structure of formula_5: there may in principle be a non-zero nilradical (intersection of all prime ideals) - and after taking the quotient by that one can speak of the product of all embeddings of "K" and "L" in various "M", "over" "N".
In case "K" and "L" are finite extensions of N, the situation is particularly simple since the tensor product is of finite dimension as an "N"-algebra (and thus an Artinian ring). One can then say that if "R" is the radical, one has formula_12 as a direct product of finitely many fields. Each such field is a representative of an equivalence class of (essentially distinct) field embeddings for "K" and "L" in some extension "M".
Examples.
For example, if "K" is generated over ℚ by the cube root of 2, then formula_13 is the product of (a copy of) "K", and a splitting field of
of degree 6 over ℚ. One can prove this by calculating the dimension of the tensor product over ℚ as 9, and observing that the splitting field does contain two (indeed three) copies of "K", and is the compositum of two of them. That incidentally shows that "R" = {0} in this case.
An example leading to a non-zero nilpotent: let
with K the field of rational functions in the indeterminate "T" over the finite field with "p" elements. (See separable polynomial: the point here is that "P" is "not" separable). If L is the field extension "K"("T"1/"p") (the splitting field of "P") then "L"/"K" is an example of a purely inseparable field extension. In formula_14 the element
is nilpotent: by taking its "p"th power one gets 0 by using "K"-linearity.
Classical theory of real and complex embeddings.
In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool. If "K" is an extension of ℚ of finite degree "n", formula_16 is always a product of fields isomorphic to ℝ or ℂ. The totally real number fields are those for which only real fields occur: in general there are "r"1 real and "r"2 complex fields, with "r"1 + 2"r"2 = "n" as one sees by counting dimensions. The field factors are in 1–1 correspondence with the "real embeddings", and "pairs of complex conjugate embeddings", described in the classical literature.
This idea applies also to formula_17 where ℚ"p" is the field of "p"-adic numbers. This is a product of finite extensions of ℚ"p", in 1–1 correspondence with the completions of K for extensions of the "p"-adic metric on ℚ.
Consequences for Galois theory.
This gives a general picture, and indeed a way of developing Galois theory
(along lines exploited in Grothendieck's Galois theory). It can be shown that for separable extensions the radical is always {0}; therefore the Galois theory case is the "semisimple" one, of products of fields alone.


Q-Charlier polynomials

In mathematics, the "q"-Charlier 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 the basic hypergeometric function by


Representation of a Lie superalgebra

In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra "L" on a Z2-graded vector space "V", such that if "A" and "B" are any two pure elements of "L" and "X" and "Y" are any two pure elements of "V", then
Equivalently, a representation of "L" is a Z2-graded representation of the universal enveloping algebra of "L" which respects the third equation above.
Unitary representation of a star Lie superalgebra.
A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and
A unitary representation of such a Lie algebra is a Z2 graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint elements of the Lie superalgebra are represented by Hermitian transformations.
This is a major concept in the study of supersymmetry together with representation of a Lie superalgebra on an algebra. Say A is an *-algebra representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L[a]*=-(-1)LaL*[a*]) and H is the unitary rep and also, H is a unitary representation of A.
These three reps are all compatible if for pure elements a in A, |ψ> in H and L in the Lie superalgebra,
Sometimes, the Lie superalgebra is embedded within A in the sense that there is a homomorphism from the universal enveloping algebra of the Lie superalgebra to A. In that case, the equation above reduces to
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers.


Quotientable automorphism

In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group.
All family automorphisms are quotientable, and particularly, all class automorphisms and power automorphisms are. As well, all inner automorphisms are quotientable, and more generally, any automorphism defined by an "algebraic formula" is quotientable.


Q-Meixner polynomials

In mathematics, the "q"-Meixner 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


Out(Fn)

In mathematics, Out("Fn") is the outer automorphism group of a free group on "n" generators. These groups play an important role in geometric group theory.
Outer space.
Out("Fn") acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.
Definition.
A point of the outer space is essentially an formula_1-graph "X" homotopy equivalent to a bouquet of "n" circles together with a certain choice of a free homotopy class of a homotopy equivalence from "X" to the bouquet of "n" circles. An formula_1-graph is just a weighted graph with weights in formula_1. The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3.
A more descriptive view avoiding the homotopy equivalence "f" is the following. We may fix an identification of the fundamental group of the bouquet of "n" circles with the free group formula_4 in "n" variables. Furthermore, we may choose a maximal tree in "X" and choose for each remaining edge a direction. We will now assign to each remaining edge "e" a word in formula_4 in the following way. Consider the closed path starting with "e" and then going back to the origin of "e" in the maximal tree. Composing this path with "f" we get a closed path in a bouquet of "n" circles and hence an element in its fundamental group formula_4. This element is not well defined; if we change "f" by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique cyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type of "f" from these data. This view has the advantage, that it avoids the extra choice of "f" and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.
The operation of Out("Fn") on the outer space is defined as follows. Every automorphism "g" of formula_4 induces a self homotopy equivalence "g′" of the bouquet of "n" circles. Composing "f" with "g′" gives the desired action. And in the other model it is just application of "g" and making the resulting word cyclically reduced.
Connection to length functions.
Every point in the outer space determines a unique length function formula_8. A word in formula_4 determines via the chosen homotopy equivalence a closed path in "X". The length of the word is then the minimal length of a path in the free homotopy class of that closed path. Such a length function is constant on each conjugacy class. The assignment formula_10 defines an embedding of the outer space to some infinite dimensional projective space.
Simplicial structure on the outer space.
In the second model an open simplex is given by all those formula_1-graphs, which have combinatorically the same underlying graph and the same edges are labeled with the same words (only the length of the edges may differ). The boundary simplices of such a simplex consists of all graphs, that arise from this graph by collapsing an edge. If that edge is a loop it cannot be collapsed without changing the homotopy type of the graph. Hence there is no boundary simplex. So one can think about the outer space as a simplicial complex with some simplices removed. It is easy to verify, that the action of formula_12 is simplicial and has finite isotropy groups.
Structure.
The abelianization map formula_13 induces a homomorphism from formula_12 to the general linear group formula_15, the latter being the automorphism group of formula_16. This map is onto, making formula_12 a group extension,
The kernel formula_19 is the Torelli group of formula_4.
In the case formula_21, the map formula_22 is an isomorphism.
Analogy with mapping class groups.
Because formula_4 is the fundamental group of a bouquet of "n" circles, formula_12 can be described topologically as the mapping class group of a bouquet of "n" circles (in the homotopy category), in analogy to the mapping class group of a closed surface which is isomorphic to the outer automorphism group of the fundamental group of that surface.


Generalized symmetric group

In mathematics, the generalized symmetric group is the wreath product formula_1 of the cyclic group of order "m" and the symmetric group of order "n".
Representation theory.
There is a natural representation of elements of formula_7 as generalized permutation matrices, where the nonzero entries are "m"-th roots of unity: formula_8
The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see .
Homology.
The first group homology group (concretely, the abelianization) is formula_9 (for "m" odd this is isomorphic to formula_10): the formula_11 factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to formula_11 (concretely, by taking the product of all the formula_11 values), while the sign map on the symmetric group yields the formula_14 These are independent, and generate the group, hence are the abelianization.
The second homology group (in classical terms, the Schur multiplier) is given by :
Note that it depends on "n" and the parity of "m:" formula_17 and formula_18 which are the Schur multipliers of the symmetric group and signed symmetric group.


Charles Leedham-Green

Charles R. Leedham-Green is a retired professor of mathematics at Queen Mary, University of London, known for his work in group theory. He completed his DPhil at the University of Oxford.
His parents were John Charles Leedham-Green (1902–1984), a surgeon and general practitioner in Southwold, and Gertrude Mary Somerville Caldwell.
Work.
With Leonard Soicher, Leedham-Green designed the product replacement algorithm; an algorithm within computational group theory that generates random elements of groups by taking a random walk through the group. This algorithm has been implemented in both GAP and MAGMA.
He is responsible for a great body of work in group theory. In recent times, this has involved research in computational group theory and pro-p groups.
The 300th edition of the "Journal of Algebra" was dedicated to him for his 65th birthday.
On the occasion of his retirement in 2006, the Mathematics Research Centre at Queen Mary held a conference in celebration of his mathematical achievements.


Jan Trlifaj

Jan Trlifaj (born 30 December 1954) is a Professor of Mathematics at Charles University whose research interests include Commutative algebra, Homological algebra and Representation theory.
Career and research.
Jan Trlifaj studied mathematics at the Faculty of Mathematics and Physics, Charles University, from which he received MSc. in 1979, Ph.D. in 1989 under Ladislav Bican. and Prof. of Mathematics in the field Algebra and number theory in 2009.
In the academic year 1994/95 he had the position as Postdoctoral Fellow of the Royal Society at Department of Mathematics at University of Manchester. In Fall 1998 he received the J.W.Fulbright Scholarship at the Department of Mathematics, University California at Irvine. During Fall 2002 and 2006 he was a visiting Professor at Centre de Recerca Matemàtica, Barcelona.
Since 1990, he has completed numerous short term visiting appointments and given over 100 invited lectures at conferences and seminars worldwide.
Since 2017, he is Fellow of Learned Society of the Czech Republic.
He served in the organizing committee of 18th International Conference on
Representations of Algebras (ICRA 2018), held for 250 participants from 34 countries in August 2018 in Prague, Czech Republic.
He has been elected Fellow of the American Mathematical Society (AMS) in 2020, for contributions to homological algebra and tilting theory for non finitely generated modules.
He serves as Member of the Science board for Neuron prize that is awarded to best Czech scientists by Neuron Endowment Fund.


Geometric quotient

In algebraic geometry, a geometric quotient of an algebraic variety "X" with the action of an algebraic group "G" is a morphism of varieties formula_1 such that
The notion appears in geometric invariant theory. (i), (ii) say that "Y" is an orbit space of "X" in topology. (iii) may also be phrased as an isomorphism of sheaves formula_7. In particular, if "X" is irreducible, then so is "Y" and formula_8: rational functions on "Y" may be viewed as invariant rational functions on "X" (i.e., rational-invariants of "X").
For example, if "H" is a closed subgroup of "G", then formula_9 is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same).
Relation to other quotients.
A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory.
A geometric quotient is precisely a good quotient whose fibers are orbits of the group.


Core (group theory)

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.
The normal core.
Definition.
For a group "G", the normal core or normal interior of a subgroup "H" is the largest normal subgroup of "G" that is contained in "H" (or equivalently, the intersection of the conjugates of "H"). More generally, the core of "H" with respect to a subset "S"⊆"G" is the intersection of the conjugates of "H" under "S", "i.e."
Under this more general definition, the normal core is the core with respect to "S"="G". The normal core of any normal subgroup is the subgroup itself.
Significance.
Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in case the action is transitive, the normal core of any isotropy subgroup is precisely the kernel of the action.
A core-free subgroup is a subgroup whose normal core is the trivial subgroup. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, faithful group action.
The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
The "p"-core.
In this section "G" will denote a finite group, though some aspects generalize to locally finite groups and to profinite groups.
Definition.
For a prime "p", the p"-core of a finite group is defined to be its largest normal p-subgroup. It is the normal core of every Sylow p-subgroup of the group. The "p"-core of "G" is often denoted formula_2, and in particular appears in one of the definitions of the Fitting subgroup of a finite group. Similarly, the p"′-core is the largest normal subgroup of "G" whose order is coprime to "p" and is denoted formula_3. In the area of finite insoluble groups, including the classification of finite simple groups, the 2′-core is often called simply the core and denoted formula_4. This causes only a small amount of confusion, because one can usually distinguish between the core of a group and the core of a subgroup within a group. The "p"′,"p"-core, denoted formula_5 is defined by formula_6. For a finite group, the "p"′,"p"-core is the unique largest normal "p"-nilpotent subgroup.
The "p"-core can also be defined as the unique largest subnormal "p"-subgroup; the "p"′-core as the unique largest subnormal "p"′-subgroup; and the "p"′,"p"-core as the unique largest subnormal "p"-nilpotent subgroup.
The "p"′ and "p"′,"p"-core begin the upper "p"-series. For sets "π"1, "π"2, ..., "π""n"+1 of primes, one defines subgroups O"π"1, "π"2, ..., "π""n"+1("G") by:
The upper "p"-series is formed by taking "π"2"i"−1 = "p"′ and "π"2"i" = "p;" there is also a lower "p"-series. A finite group is said to be p"-nilpotent if and only if it is equal to its own "p"′,"p"-core. A finite group is said to be p"-soluble if and only if it is equal to some term of its upper "p"-series; its "p"-length is the length of its upper "p"-series. A finite group "G" is said to be p-constrained for a prime "p" if formula_8.
Every nilpotent group is "p"-nilpotent, and every "p"-nilpotent group is "p"-soluble. Every soluble group is "p"-soluble, and every "p"-soluble group is "p"-constrained. A group is "p"-nilpotent if and only if it has a normal "p"-complement, which is just its "p"′-core.
Significance.
Just as normal cores are important for group actions on sets, "p"-cores and "p"′-cores are important in modular representation theory, which studies the actions of groups on vector spaces. The "p"-core of a finite group is the intersection of the kernels of the irreducible representations over any field of characteristic "p". For a finite group, the "p"′-core is the intersection of the kernels of the ordinary (complex) irreducible representations that lie in the principal "p"-block. For a finite group, the "p"′,"p"-core is the intersection of the kernels of the irreducible representations in the principal "p"-block over any field of characteristic "p". Also, for a finite group, the "p"′,"p"-core is the intersection of the centralizers of the abelian chief factors whose order is divisible by "p" (all of which are irreducible representations over a field of size "p" lying in the principal block). For a finite, "p"-constrained group, an irreducible module over a field of characteristic "p" lies in the principal block if and only if the "p"′-core of the group is contained in the kernel of the representation.
Solvable radicals.
A related subgroup in concept and notation is the solvable radical. The solvable radical is defined to be the largest solvable normal subgroup, and is denoted formula_9. There is some variance in the literature in defining the "p"′-core of "G". A few authors in only a few papers (for instance Thompson's N-group papers, but not his later work) define the "p"′-core of an insoluble group "G" as the "p"′-core of its solvable radical in order to better mimic properties of the "2"′-core.


N-group (finite group theory)

In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial "p"-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.
Simple N-groups.
The simple N-groups were classified by in a series of 6 papers totaling about 400 pages.
The simple N-groups consist of the special linear groups PSL2("q"), PSL3(3), the Suzuki groups Sz(22"n"+1), the unitary group U3(3), the alternating group "A"7, the Mathieu group M11, and the Tits group. (The Tits group was overlooked in Thomson's original announcement in 1968, but Hearn pointed out that it was also a simple N-group.) More generally Thompson showed that any non-solvable N-group is a subgroup of Aut("G") containing "G" for some simple N-group "G".
 generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable. The only extra simple groups that appear are the unitary groups U3("q").
Proof.
 gives a summary of Thompson's classification of N-groups.
The primes dividing the order of the group are divided into four classes π1, π2, π3, π4 as follows
The proof is subdivided into several cases depending on which of these four classes the prime 2 belongs to, and also on an integer "e", which is the largest integer for which there is an elementary abelian subgroup of rank "e" normalized by a nontrivial 2-subgroup intersecting it trivially.
Consequences.
A minimal simple group is a non-cyclic simple group all of whose proper subgroups are solvable.
The complete list of minimal finite simple groups is given as follows
In other words a non-cyclic finite simple group must have a subquotient isomorphic to one of these groups.


Seminormal subgroup

In mathematics, in the field of group theory, a subgroup formula_1 of a group formula_2 is termed seminormal if there is a subgroup formula_3 such that formula_4, and for any proper subgroup formula_5 of formula_3, formula_7 is a proper subgroup of formula_2.
This definition of seminormal subgroups is due to Xiang Ying Su.
Every normal subgroup is seminormal. For finite groups, every quasinormal subgroup is seminormal.


Semisimple representation

In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity.
Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group "G" over a field k is a semisimple module over the group ring "k"["G"].
Equivalent characterizations.
Let "V" be a representation of a group "G"; or more generally, let "V" be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple representations in that sense.
The following are equivalent:
The equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:
Examples and non-examples.
Unitary representations.
A finite-dimensional unitary representation (i.e., a representation factoring through a unitary group) is a basic example of a semisimple representation. Such a representation is semisimple since if "W" is a subrepresentation, then the orthogonal complement to "W" is a complementary representation because if formula_2 and formula_3, then formula_4 for any "w" in "W" since "W" is "G"-invariant, and so formula_5.
For example, given a continuous finite-dimensional complex representation formula_6 of a finite group or a compact group "G", by the averaging argument, one can define an inner product formula_7 on "V" that is "G"-invariant: i.e., formula_8, which is to say formula_9 is a unitary operator and so formula_10 is a unitary representation. Hence, every finite-dimensional continuous complex representation of "G" is semisimple. For a finite group, this is a special case of Maschke's theorem, which says a finite-dimensional representation of a finite group "G" over a field k with characteristic not dividing the order of "G" is semisimple.
Representations of semisimple Lie algebras.
By Weyl's theorem on complete reducibility, every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is semisimple.
Separable minimal polynomials.
Given a linear endomorphism "T" of a vector space "V", "V" is semisimple as a representation of "T" (i.e., "T" is a semisimple operator) if and only if the minimal polynomial of "T" is separable; i.e., a product of distinct irreducible polynomials.
Associated semisimple representation.
Given a finite-dimensional representation "V", the Jordan–Hölder theorem says there is a filtration by subrepresentations: formula_11 such that each successive quotient formula_12 is a simple representation. Then the associated vector space formula_13 is a semisimple representation called an associated semisimple representation, which, up to an isomorphism, is uniquely determined by "V".
Unipotent group non-example.
A representation of a unipotent group is generally not semisimple. Take formula_14 to be the group consisting of real matrices formula_15; it acts on formula_16 in a natural way and makes "V" a representation of "G". If "W" is a subrepresentation of "V" that has dimension 1, then a simple calculation shows that it must be spanned by the vector formula_17. That is, there are exactly three "G"-subrepresentations of "V"; in particular, "V" is not semisimple (as a unique one-dimensional subrepresentation does not admit a complementary representation).
Semisimple decomposition and multiplicity.
The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example, for a trivial representation, simple representations are one-dimensional vector spaces and thus a semisimple decomposition amounts to a choice of a basis of the representation vector space. The isotypic decomposition, on the other hand, is an example of a unique decomposition.
However, for a finite-dimensional semisimple representation "V" over an algebraically closed field, the numbers of simple representations up to isomorphisms appearing in the decomposition of "V" (1) are unique and (2) completely determine the representation up to isomorphisms; this is a consequence of Schur's lemma in the following way. Suppose a finite-dimensional semisimple representation "V" over an algebraically closed field is given: by definition, it is a direct sum of simple representations. By grouping together simple representations in the decomposition that are isomorphic to each other, up to an isomorphism, one finds a decomposition (not necessarily unique):
where formula_19 are simple representations, mutually non-isomorphic to one another, and formula_20 are positive integers. By Schur's lemma,
where formula_22 refers to the equivariant linear maps. Also, each formula_20 is unchanged if formula_19 is replaced by another simple representation isomorphic to formula_19. Thus, the integers formula_20 are independent of chosen decompositions; they are the "multiplicities" of simple representations formula_19, up to isomorphisms, in "V".
In general, given a finite-dimensional representation formula_6 of a group "G" over a field "k", the composition formula_29 is called the character of formula_30. When formula_30 is semisimple with the decomposition formula_18 as above, the trace formula_33 is the sum of the traces of formula_34 with multiplicities and thus, as functions on "G",
where formula_36 are the characters of formula_19. When "G" is a finite group or more generally a compact group and formula_38 is a unitary representation with the inner product given by the averaging argument, the Schur orthogonality relations say: the irreducible characters (characters of simple representations) of "G" are an orthonormal subset of the space of complex-valued functions on "G" and thus formula_39.
Isotypic decomposition.
There is a decomposition of a semisimple representation that is unique, called "the" isotypic decomposition of the representation. By definition, given a simple representation "S", the isotypic component of type "S" of a representation "V" is the sum of all subrepresentations of "V" that are isomorphic to "S"; note the component is also isomorphic to the direct sum of some choice of subrepresentations isomorphic to "S" (so the component is unique, while the summands are not necessary so).
Then the isotypic decomposition of a semisimple representation "V" is the (unique) direct sum decomposition:
where formula_41 is the set of isomorphism classes of simple representations of "V" and formula_42 is the isotypic component of "V" of type "S" for some formula_43.
Example.
Let formula_38 be the space of homogeneous degree-three polynomials over the complex numbers in variables formula_45. Then formula_46 acts on formula_38 by permutation of the three variables. This is a finite-dimensional complex representation of a finite group, and so is semisimple. Therefore, this 10-dimensional representation can be broken up into three isotypic components, each corresponding to one of the three irreducible representations of formula_46. In particular, formula_38 contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation formula_50 of formula_46. For example, the span of formula_52 and formula_53 is isomorphic to formula_50. This can more easily be seen by writing this two-dimensional subspace as
Another copy of formula_50 can be written in a similar form:
So can the third:
Then formula_59 is the isotypic component of type formula_50 in formula_38.
Completion.
In Fourier analysis, one decomposes a (nice) function as the "limit" of the Fourier series of the function. In much the same way, a representation itself may not be semisimple but it may be the completion (in a suitable sense) of a semisimple representation. The most basic case of this is the Peter–Weyl theorem, which decomposes the left (or right) regular representation of a compact group into the Hilbert-space completion of the direct sum of all simple unitary representations. As a corollary, there is a natural decomposition for formula_62 = the Hilbert space of (classes of) square-integrable functions on a compact group "G":
where formula_64 means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations formula_30 of "G". Note here that every simple unitary representation (up to an isomorphism) appears in the sum with the multiplicity the dimension of the representation.
When the group "G" is a finite group, the vector space formula_66 is simply the group algebra of "G" and also the completion is vacuous. Thus, the theorem simply says that
That is, each simple representation of "G" appears in the regular representation with multiplicity the dimension of the representation. This is one of standard facts in the representation theory of a finite group (and is much easier to prove).
When the group "G" is the circle group formula_68, the theorem exactly amounts to the classical Fourier analysis.
Applications to physics.
In quantum mechanics and particle physics, the angular momentum of an object can be described by , all of which are semisimple. Due to connection between SO(3) and SU(2), the non-relativistic spin of an elementary particle is described by complex representations of SU(2) and the relativistic spin is described by complex representations of SL2(C), all of which are semisimple. In angular momentum coupling, Clebsch–Gordan coefficients arise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.


Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
Definitions.
Via the hypergeometric function.
The Jacobi polynomials are defined via the hypergeometric function as follows:
where formula_2 is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:
Rodrigues' formula.
An equivalent definition is given by Rodrigues' formula:
If formula_5, then it reduces to the Legendre polynomials:
Alternate expression for real argument.
For real "x" the Jacobi polynomial can alternatively be written as
and for integer "n"
where is the Gamma function.
In the special case that the four quantities , and are nonnegative integers, the Jacobi polynomial can be written as
The sum extends over all integer values of "s" for which the arguments of the factorials are nonnegative.
Basic properties.
Orthogonality.
The Jacobi polynomials satisfy the orthogonality condition
As defined, they do not have unit norm with respect to the weight. This can be corrected by dividing by the square root of the right hand side of the equation above, when formula_13.
Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:
Symmetry relation.
The polynomials have the symmetry relation
thus the other terminal value is
Derivatives.
The "k"th derivative of the explicit expression leads to
Differential equation.
The Jacobi polynomial is a solution of the second order linear homogeneous differential equation
Recurrence relations.
The recurrence relation for the Jacobi polynomials of fixed "α","β" is:
for "n" = 2, 3, ...
Since the Jacobi polynomials can be described in terms of the hypergeometric function, recurrences of the hypergeometric function give equivalent recurrences of the Jacobi polynomials. In particular, Gauss' contiguous relations correspond to the identities
Generating function.
The generating function of the Jacobi polynomials is given by
where
and the branch of square root is chosen so that "R"("z", 0) = 1.
Asymptotics of Jacobi polynomials.
For "x" in the interior of , the asymptotics of for large "n" is given by the Darboux formula
where
and the "O" term is uniform on the interval [ε, -ε] for every ε > 0.
The asymptotics of the Jacobi polynomials near the points ±1 is given by the Mehler–Heine formula
where the limits are uniform for "z" in a bounded domain.
The asymptotics outside is less explicit.
Applications.
Wigner d-matrix.
The expression () allows the expression of the Wigner d-matrix "d""j""m"’,"m"(φ) (for 0 ≤ φ ≤ 4) in terms of Jacobi polynomials: