1 Representation & FG-homomorphism

Basic Definitions

Definition

Consider a homomorphism from a group to a general linear group:

$$\rho :G\to \mathrm{GL}(n,F),$$

and we call $\rho$ is a representation of group $G$.

Definition

An $FG$-module is a vector space $V$ over field $F$, which satisfies the following properties:

• $vg\in V$
• associativity: $(vg)h=v(gh)$
• identity axiom: $v1=v$
• $(\lambda v)g=\lambda (vg)$
• $(u+v)g=ug+vg$

Remark. For a $FG$-module and a fixed basis, we get a unique group representation. If the basis is changed, we get similar matrices. Conversely, from a group representation we can get conjugate transformations.

Definition

We say $\theta$ is a $FG$-homomorphism, if $\theta$ is a linear transformation and

$$\theta :V\to W,(v\theta )g=(vg)\theta .$$

In the other word, if $\theta$ sends $v$ to $w$, then it sends $vg$ to $wg$.

Remark. Note that $FG$-module is a category, where object = vector, morphism = element of group, and they satisfy associativity and identity axiom. In addition, $FG$-homomorphism can be viewed as a functor between two $FG$-modules.

Definition

A submodule is a subspace and invariant under any element $g$ of $G$. In the other word, $V⩽U$ is a submodule, if $vg\in V$ for all $v\in V$ and $g\in G$. A $FG$-module is irreducible if it has no non-trivial $FG$-submodule.

An $FG$-module $V$ is faithful if the identity element of $G$ is the only element $g$ for which $vg=v$ for all $v\in V$.