代数学笔记——有限群的表示论（一）

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这个专题主要介绍有限群的线性表示。粗略地说，我们把有限群中的元素看作是某个有限维线性空间上的（可逆）线性变换，从而来研究群的性质。本文主要介绍群表示论的最基础的知识，并介绍 Maschke 定理，这个定理在群表示论的教材中往往一开始就会碰到，虽然证明并不难，但它在有限群的表示论中起着至关重要的作用。

Let $V$ be a finite dimensional vector space over a field $k$ . Let $G$ be a finite group. A linear representation of $G$ in $V$ is a group homomorphism $\rho:G\to\operatorname{GL}(V)$ .

Define the group algebra $k[G]=\{\sum_{g\in G}a_g g:a_g\in k\}$ with obvious addition and multiplication. Then linear representations of $G$ are the same things as $k[G]$ -modules, since any group homomorphism $G\to\operatorname{GL}(V)$ can be linearly extended to a $k$ -algebra homomorphism $k[G]\to\operatorname{End}_k (V)$ .

Let $\rho:G\to\operatorname{GL}(V)$ , $\rho':G\to\operatorname{GL}(V')$ be two representations of $G$ . We say a linear map $\varphi:V\to V'$ is a morphism of representations of $G$ if the diagram

commutes for all $g\in G$ . This exactly means that $\varphi$ is a $k[G]$ -module homomorphism.

Remark

In fact a representation of $G$ can be viewed as a functor $\rho$ from the category $\mathcal{C}$ consisting of a single object $X$ whose morphisms are elements of the group $G$ , to the category of $k$ -vector spaces, sending a group element $g$ to $\rho(g)\in\operatorname{GL}(V)$ . A natural transformation from $\rho$ to $\rho'$ is exactly a linear map $\varphi:V\to V'$ such that the diagram above commutes, i.e. a morphism of representations is exactly a natural transformation. Then we say two representations $\rho$ and $\rho'$ of $G$ are isomorphic if they are naturally isomorphic as functors.

Definition

We say that a representation $\rho:G\to\operatorname{GL}(V)$ is simple (or irreducible) if $V$ has no $k[G]$ -submodules except $0,V$ .

Note that there are only finitely many irreducible representations up to isomorphism. This is because any irreducible $k[G]$ -module is isomorphic to a quotient module of $V$ , thus is a composition factor of $V$ , and $V$ has finitely many composition factors.

If $\rho_1:G\to \operatorname{GL}(V_1)$ , $\rho_2:G\to \operatorname{GL}(V_2)$ are two representations of $G$ , then we can define their

(1) direct sum

$\rho_1\oplus\rho_2:G\to\operatorname{GL}(V_1\oplus V_2)$ , $(\rho_1\oplus\rho_2)(g)(v_1,v_2)= (\rho_1(g)v_1,\rho_2(g)v_2)$ .

(2) tensor product

$\rho_1\otimes\rho_2:G\to\operatorname{GL}(V_1\otimes V_2)$ , $(\rho_1\otimes\rho_2)(g)(v_1\otimes v_2) = \rho_1(g)v_1\otimes \rho_2(g)v_2$

Suppose that $\rho_1:G\to \operatorname{GL}(V_1)$ , $\rho_2:G\to \operatorname{GL}(V_2)$ are two representations of $G$ . For any linear map $f:V_1\to V_2$ , define its trace to be

$\operatorname{Tr}_G(f):V_1\to V_2, \quad v\mapsto\sum_{g\in G}\rho_2(g)\circ f\circ \rho_1(g^{-1})v.$

Then $\operatorname{Tr}_G(f)$ is a morphism of $k[G]$ -modules.

Suppose $\rho_1$ , $\rho_2$ are irreducible representations. By Schur's lemma, if $\rho_1$ and $\rho_2$ are not isomorphic, then $\operatorname{Tr}_G(f)=0$ . If $k=\mathbb{C}$ , and $\rho_1=\rho_2:G\to\operatorname{GL}(V)$ , then $\operatorname{Tr}_G(f)=\lambda\cdot\operatorname{id}_V$ with $\lambda=\frac{|G|}{\dim V}\operatorname{tr}(f)$ (by taking trace).

Theorem[Maschke]

The group algebra $k[G]$ is semisimple if and only if $\operatorname{char} (k)\nmid |G|$ .

Proof

Suppose $\operatorname{char}(k)\nmid |G|$ . Then for any representation $\rho:G\to \operatorname{GL}(V)$ , and any $k[G]$ -submodule $W$ of $V$ , let $W'$ be any subspace of $V$ such that $V=W\oplus W'$ . Let $\pi:V\to W$ be the projection map with respect to this composition. Then $\psi:=\frac{1}{|G|}\operatorname{Tr}_G(\pi)$ is a morphism of $k[G]$ -modules and we have $\psi|_W=\operatorname{id}_W$ . Thus the exact sequence of $k[G]$ -modules

splits. Hence $V\cong W\oplus\ker\psi$ as $k[G]$ -modules. Therefore, $k[G]$ is semisimple.

On the other hand, if $\operatorname{char}(k)\mid |G|$ , then $(\sum_{g\in G} g)^2=|G|\sum_{g\in G} g=0$ . Hence $\sum_{g\in G} g$ is contained in the nilradical of $k[G]$ . Thus the Jacobson radical is not 0. So $k[G]$ is not semisimple.
$\square$

Remark

Note that if $k[G]$ is semisimple, then $\rho$ is irreducible if and only if $\rho$ is indecomposable. In this case, every representation of $G$ is a direct sum of irreducible representations, by induction on $\dim V$ .

编辑于 2020-10-13

代数学笔记——有限群的表示论（一）

Remark

Definition

Theorem[Maschke]

Proof

Remark

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