Representation ring - Knowlpedia

Definition

Let $G$ be a group, topological group, or affine group scheme, and fix a coefficient field $k$ . The representation ring $R_k(G)$ is the Grothendieck group of the exact category of finite-dimensional $k$ -representations of $G$ , with multiplication induced by tensor product:

[V]+[W]=[V\oplus W], \qquad [V][W]=[V\otimes_k W].

Equivalently, the additive relations are $[V]=[V']+[V'']$ for every short exact sequence $0\to V'\to V\to V''\to0$ .

Semisimple case

If the representation category is semisimple, $R_k(G)$ is the free abelian group on the irreducible representations. Dual representations define an involution $[V]\mapsto[V^\vee]$ , and exterior powers give it a $\lambda$ -ring structure.

Characters

Taking traces defines a ring homomorphism from $R_k(G)$ to class functions. For finite groups in characteristic $0$ , or for complex reductive groups in the appropriate regular-character setting, this character map is injective. The statement that characters separate semisimple conjugacy classes concerns the resulting functions on the semisimple quotient, not arbitrary elements in all characteristics.

References

Jean-Pierre Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer, 1977, Chapters 2 and 9.
Alexander Grothendieck, “La théorie des classes de Chern,” Bulletin de la Société Mathématique de France 86 (1958), 137–154. Numdam.