What algebraic structure do the representations of a fixed finite group carry? The set of isomorphism classes of finite-dimensional complex representations of has two natural operations: direct sum and tensor product . These are commutative, associative, and tensor distributes over direct sum: . This is almost a ring, except that we lack additive inverses.
Definition 14.2.1 (Representation ring). Let be the free abelian group on isomorphism classes of finite-dimensional complex representations of , i.e. formal -linear combinations . Define to be the quotient of by the subgroup generated by all elements . Then is a commutative ring with:
Addition: (and now we can subtract: makes sense).
Multiplication: .
Identity: , the class of the trivial representation.
This is the representation ring (or Green ring) of .
By complete reducibility and uniqueness of decomposition into irreducibles, every element of can be written uniquely as where are the irreducible representations and . So as an abelian group.
General elements of with some are called virtual representations. The actual representations (those with all ) form a cone inside .
Proposition 14.2.1 (The character map is a ring homomorphism). The character map extends by linearity to a ring homomorphism where denotes the ring of class functions under pointwise addition and multiplication. This is a ring homomorphism because and .
The image of is the lattice of virtual characters . After extending scalars, our basis theorem gives:
Corollary 14.2.1. The induced map is an isomorphism of -algebras.
Note. There are theorems of Artin and Brauer that describe the lattice of virtual characters inside :
(Artin) Every character of a representation of is a -linear combination of characters induced from cyclic subgroups of .
(Brauer) Every character of a representation of is a -linear combination of characters of 1-dimensional representations induced from “elementary” subgroups of , where an elementary subgroup is a product with cyclic of order coprime to and a -group, for some prime .
See Serre, Linear Representations of Finite Groups, for proofs.