Definition 1

Let M be a K-module. A linear representation of G in M is a group homomorphism from G to the linear group $\mathbf{G}\mathbf{L}(M)$ (II, §1, No. 2, p. 195).

Let $\pi$ be a linear representation of G in a K-module M. We also say that the pair $(M, \pi )$ is a linear representation of G. When K is nonzero and M is a free K-module, the dimension of M is called the degree (or dimension) of the representation $\pi$.

Recall that the canonical mapping $g\mapsto e_g$ from G to the algebra K[G] of the group G is a basis of the K-module K[G]. By abuse of notation, we also denote the element $e_g$ of K[G] by $g$. Given a linear representation $\pi : G\rightarrow \mathbf{G}\mathbf{L}(M)$ of G, there exists a unique homomorphism of K-algebras from K[G] to End$_K(M)$ that extends $\pi$ (III, §2, No. 6, p. 447, Example); we also denote this extension by $\pi$. It is a representation of the algebra K[G] in M (VIII, p. 373, Definition 1), so that M is a K[G]-module. Conversely, every linear representation $\rho$ of the algebra K[G] in M defines a linear representation of G in M given by $g\mapsto \rho (g)$.

We will freely apply the terminology concerning the K[G]-module structure to linear representations of the group G. For example, we speak of a subrepresentation, a simple representation, a direct sum of representations, etc. Let $(M, \pi )$ and $(M', \pi ')$ be linear representations of G; we denote by Hom$_G(\pi , \pi ')$ the K-module Hom$_{K[G]}(M,M')$ of homomorphisms of K[G]-modules from M to $M'$.[^1]

A mapping $f$ from G to K is called a central function if we have $f(gg') =$ $f(g'g)$ for every pair $(g, g')$ of elements of G; equivalently, we have $f(ghg^{-1}) =$ $f(h)$ for every pair of elements $g, h$ of G. The central functions are therefore the functions on G whose restriction to each conjugacy class is constant. They form a submodule of the space of mappings from G to K, denoted by $\mathscr{Z}_K(G)$. When G is finite, the K-algebra $\mathscr{Z}_K(G)$ is a free K-module of dimension the number of conjugacy classes of G. The center Z(K[G]) of the algebra K[G] consists of the elements $a=\sum a_gg$ of K[G] such that $hah^{-1}=a$ for every $h\in G$. Now, we have

consequently, when G is finite, the center Z(K[G]) of the algebra K[G] consists of the central functions.

Let $(M, \pi )$ be a linear representation of G. Suppose that M is a finite-dimensional free K-module. The trace of $\pi$ is the trace of the representation of K[G] associated with $\pi$, that is (VIII, p. 382), the linear form $a\mapsto$ Tr($\pi (a)$) on K[G]. This linear form is determined by the mapping $g\mapsto$ Tr($\pi (g)$) from G to K, which we call the character of the representation $\pi$ and denote by $\chi_{\pi}$. The character of a representation is a central function (II, §4, No. 3, p. 273, Proposition 3).

Let M and $M'$ be finite-dimensional free K-modules. Let $\pi$ and $\pi '$ be linear representations of G in M and $M'$, respectively. Then $M\oplus M'$ is a finite-dimensional free K-module, and, by Proposition 1 of III, §9, No. 2, p. 543, we have

More generally, let

be an exact sequence of K[G] modules, and let $\pi ,\pi '$, and $\pi ''$ be the linear representations of G associated with $M, M'$, and $M''$, respectively. Suppose that the K-modules $M'$ and $M''$ are free and finite-dimensional. Then so is M (II, §1, No. 11, p. 218, Proposition 21), and we have