Group algebra

Definition

Let $G$ be a finite group and $k$ a field. The group algebra $k[G]$ (also written $kG$) is the $k$-vector space with basis $\{\,\delta_g : g\in G\,\}$ and multiplication determined by

extended $k$-bilinearly. Thus every element has a unique expression

and multiplication is

The identity element of $k[G]$ is $\delta_e$, where $e$ is the identity of $G$.

Representations as modules

A (finite-dimensional) group representation $\rho:G\to \mathrm{GL}(V)$ on a $k$-vector space $V$ extends uniquely to a $k$-algebra homomorphism

Equivalently, giving a representation of $G$ is the same as giving a left $k[G]$-module structure on $V$ (i.e. an action $k[G]\times V\to V$ that is $k$-bilinear and associative). In this correspondence:

• subrepresentations are exactly $k[G]$-submodules,
• irreducible representations are exactly simple modules over $k[G]$,
• complete reducibility is a statement about $k[G]$ being semisimple (cf. Maschke’s theorem and semisimple modules ).

Examples

Example 1: Cyclic groups $C_n$

Let $G=C_n=\langle t\mid t^n=e\rangle$. Then

via $\delta_{t^m}\mapsto t^m$. This realizes $k[C_n]$ as a commutative $k$-algebra.

Example 2: The order-2 group $C_2=\{e,s\}$

Here $k[C_2]=k\delta_e\oplus k\delta_s$ with $\delta_s^2=\delta_e$. So

If $\mathrm{char}(k)\neq 2$, the elements

satisfy $e_\pm^2=e_\pm$ and $e_+e_-=0$, giving a decomposition $k[C_2]\cong k\times k$. (This is a concrete instance of semisimplicity in characteristic not dividing $|G|$.)

Example 3: $S_3$ and class sums

For $G=S_3$, $k[S_3]$ is $6$-dimensional with basis $\{\delta_\sigma:\sigma\in S_3\}$. The center $Z(k[S_3])$ is spanned by sums over conjugacy classes :

These “class sums” act as scalars in any irreducible representation (compare Schur’s lemma ).