Representation ring
Encyclopedia
In mathematics
, especially in the area of algebra
known as representation theory
, the representation ring of a group
is a ring
formed from all the (isomorphism classes of the) linear representations
of the group. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed field
s of characteristic p where the Sylow p-subgroups are cyclic
is also theoretically approachable.
of representations, and multiplication by their tensor product
over F. When F is omitted from the notation, as in R(G), then F is implicitly taken to be the field of complex numbers.
χ:G → C. Such a function is constant on conjugacy classes of G, a so-called class function
; denote the ring of class functions by C(G). The homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from RF(G) → C(G) defined by Brauer characters
is no longer injective.
For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, the representation ring of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
formed from all the (isomorphism classes of the) linear representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of the group. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
s of characteristic p where the Sylow p-subgroups are cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
is also theoretically approachable.
Formal definition
Given a group G and a field F, the elements of its representation ring RF(G) are the formal differences of isomorphism classes of finite dimensional linear F-representations of G. For the ring structure, addition is given by the Cartesian productDirect product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of representations, and multiplication by their tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
over F. When F is omitted from the notation, as in R(G), then F is implicitly taken to be the field of complex numbers.
Examples
- For the complex representations of the cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order n, the representation ring RC(Cn) is isomorphic to Z[X]/(Xn − 1), where X corresponds to the complex representation sending a generator of the group to a primitive nth root of unity. - For the rational representations of the cyclic group of order 3, the representation ring RQ(C3) is isomorphic to Z[X]/(X2 − X − 2), where X corresponds to the irreducible rational representation of dimension 2.
- For the modular representations of the cyclic group of order 3 over a field F of characteristic 3, the representation ring RF(C3) is isomorphic to Z[X,Y]/(X2 − Y − 1, XY − 2Y,Y2 − 3Y).
- The ring R(S1) for the circle group is isomorphic to Z[X, X −1]. The ring of real representations is the subring of R(G) of elements fixed by the involution on R(G) given by X → X −1.
- The ring RC(S3) for the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on three points is isomorphic to Z[X,Y]/(XY − Y,X2 − 1,Y2 − X − Y − 1), where X is the 1-dimensional alternating representation and Y the 2-dimensional irreducible representation of S3.
Characters
Any representation defines a characterCharacter theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
χ:G → C. Such a function is constant on conjugacy classes of G, a so-called class function
Class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...
; denote the ring of class functions by C(G). The homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from RF(G) → C(G) defined by Brauer characters
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
is no longer injective.
For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).