Representation ring

Formal definition

Given a group G and a field F, the elements of its representation ring R**F(G) are the formal differences of isomorphism classes of finite-dimensional F-representations of G. For the ring structure, addition is given by the direct sum 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.

The representation ring of G is the Grothendieck ring of the category of finite-dimensional representations of G.

Examples

• For the complex representations of the cyclic group of order n, the representation ring RC(C**n) is isomorphic to Z[X]/(X**n − 1), where X corresponds to the complex representation sending a generator of the group to a primitive nth root of unity.
• More generally, the complex representation ring of a finite abelian group may be identified with the group ring of the character group.
• 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 R**F(C3) is isomorphic to Z[X,Y]/(X 2 − Y − 1, XY − 2Y,Y 2 − 3Y).
• The continuous representation 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 group of degree three is isomorphic to Z[X,Y]/(XY − Y,X 2 − 1,Y 2 − X − Y − 1), where X is the 1-dimensional alternating representation and Y the 2-dimensional irreducible representation of S3.

Characters

Any finite-dimensional complex representation ρ of a group G defines a function χ:G → \mathbb{C} by the formula χ(g) = tr(ρ(g)). Such a function is a so-called class function, meaning that it is constant on each conjugacy class of G. Denote the ring of complex-valued class functions by C(G). The map sending isomorphism classes of representations to their characters gives a homomorphism R(G) → C(G), and when G is finite this is injective, so that R(G) can be identified with a subring of C(G).

In the case of finite groups this ring homomorphism R(G) → C(G) extends to an algebra isomorphism \mathbb{C} \otimes_{\mathbb{Z}} R(G) → C(G). Since isomorphism classes of irreducible representations of a finite group form a basis of \mathbb{C} \otimes_{\mathbb{Z}} R(G), while characteristic functions of conjugacy classes form a basis of C(G), this shows that a finite group has as many isomorphism classes of irreducible representations as it has conjugacy classes.

For a compact connected Lie 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).

λ-ring and Adams operations

Given a representation of G and a natural number n, we can form the n-th exterior power of the representation, which is again a representation of G. This induces an operation λn : R(G) → R(G). With these operations, R(G) becomes a λ-ring.

The Adams operations on the representation ring R(G) are maps Ψk characterised by their effect on characters χ: :\Psi^k \chi (g) = \chi(g^k) \ .

The operations Ψk are ring homomorphisms of R(G) to itself, and on representations ρ of dimension d

:\Psi^k (\rho) = N_k(\Lambda^1\rho,\Lambda^2\rho,\ldots,\Lambda^d\rho) \

where the Λi**ρ are the exterior powers of ρ and N**k is the k-th power sum expressed as a function of the d elementary symmetric functions of d variables.

References

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References

1. https://math.berkeley.edu/~teleman/math/RepThry.pdf, page 20
2. Serre, Jean-Pierre. (1977). "Linear Representations of Finite Groups". Springer New York.