群表示论 复习速览

group representation

Def. representation A representation of $G$ over $F$ is the homomorphism $\rho$ \(\rho : G \to GL(n,F)\) the degree of $\rho$ is the integer $n$

Def. equivalent Let $\rho : G \to GL(m,F)$ and $\sigma : G \to GL(n,F)$ be representations of $G$ over $F$. we say $\rho$ is equivalent to $\sigma$ if $n=m$ and there exists an invertible $n \times n$ matrix $T$ such that for all $g \in G$, \(g \sigma= T^{-1}(g \rho)T\)

也就是只是换了一组基的关系。

Def. $\rho$ is faithful if $\mathrm{Ker}\rho = {1}$.

$FG$-module

Def. $V$ a vector space over $F$ and $G$ is a group. $V$ is a $FG$-module if a multiplication $vg(v \in V,g\in G)$ is defined, satisfying the following conditions for all $u,v \in V, \lambda \in F$ and $g,h \in G$: 1)$vg \in V$ 2)$v(gh) = (vg)h$ 3)$v1=v$ 4)$(\lambda v) g = \lambda (vg)$ 5)$(u+v)g=ug+vg$

Every representation gives out a $FG$-module

Def. $G$ is a subgroup of $S_n$. The $FG$-module $V$ with basis $v_1, \dots, v_n$ such that \(v_ig=v_{ig}\text{ for all $i$ and all $g \in G$}\) if called the permutation module for $G$ over $F$.

irreducible $FG$-module

Def. $FG$-submodule $V$ an $FG$-module. A subset $W$ of $V$ is said to be an $FG$-submodule of $V$ if $W$ is a subspace and $wg \in W$ for all $w \in W$ and all $g \in G$.

Def. irreducible An $FG$-module $V$ is said to be irreducible if it is non-zero and it has no $FG$-submodules apart from ${0}$ and $V$.

group algebra and regular representation

Def. group algebra The cector space $FG$, with multiplication difined by \(\left(\sum_{g \in G } \lambda_g g \right) \left(\sum_{h \in G } \mu_h h \right) = \sum_{g,h \in G} \lambda_g \mu _h(gh)\) $\lambda_g, \mu _h \in F$

Def. homomorphism $V$ $W$ are $FG$-modules. A function $f:V \to W$ is said to be an $FG$-homomorphism if $f$ is a linear transformation and \((vg)f = (vf)g \text{ for all } v \in V, g \in G\)

If $f$ is inversable, then $f$ is an isomorphism.

Maschke’s Theorem

Thm. $G$ finite group, let $F$ be $\mathbb R$ or $\mathbb C$, and let $V$ be an $FG$-module. If $U$ is an $FG$-submodule of $V$, then there is an $FG$-submodule $W$ of $V$ such that \(V = U \oplus W\)

Schur’s lemma

Thm. Let $V$ and $W$ be irreducible $\mathbb C G$-modules. 1)If $f:V \to W$ is a $\mathbb C G$-homomorphism, then either $f$ is $\mathbb C G$-isomorphism, or $v f = 0$ for all $v \in V$. 2)If $f:V \to V$ is a $\mathbb C G$-isomorphism, then $f$ is a scalar multiple of the identity endomorphism $1_V$.

Cor. If $\rho : G \to GL(n,\mathbb C)$ is a represetation, $\rho$ is irreducible iff every $n \times n$ matrix $A$ which satisfies \((g \rho) A = A (g\rho) \text{ for all } g \in G\) has the form $A = \lambda I_n$ with $\lambda \in \mathbb C$.