Group algebra - Knowlpedia

Let $G$ be a group and $R$ a commutative ring with identity. The group ring $R[G]$ is the free $R$ -module with basis $\{\delta_g:g\in G\}$ and multiplication determined by

\delta_g\cdot \delta_h = \delta_{gh}\quad (g,h\in G),

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

x=\sum_{g\in G} a_g\,\delta_g\qquad (a_g\in R),

with only finitely many nonzero coefficients, and multiplication is

\left(\sum_{g} a_g\delta_g\right)\left(\sum_{h} b_h\delta_h\right)=\sum_{g,h} a_g b_h\,\delta_{gh}.

The identity is $\delta_e$ , where $e$ is the identity of $G$ . When $R=k$ is a field, $k[G]$ is called the group algebra (also written $kG$ ). Neither construction requires $G$ to be finite.

Representations as modules

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

\widetilde{\rho}:k[G]\to \mathrm{End}_k(V),\qquad \widetilde{\rho}\!\left(\sum_g a_g\delta_g\right)=\sum_g a_g\,\rho(g).

Equivalently, giving a representation of $G$ is the same as giving a unital left $k[G]$ -module structure on $V$ extending its scalar action. In this correspondence:

subrepresentations are exactly $k[G]$ -submodules,
irreducible representations are exactly simple modules over $k[G]$ ,
for finite $G$ , Maschke’s theorem describes when $k[G]$ is semisimple and its representations are completely reducible.

Integral orders

For finite $G$ , $\mathbb Z[G]$ is a $\mathbb Z$ -order in $\mathbb Q[G]$ : its basis has $|G|$ elements, it spans $\mathbb Q[G]$ , and its multiplication has integer structure constants. For infinite $G$ , it is not a finite-rank order in this sense.

Examples

Example 1: Cyclic groups $C_n$

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

k[C_n]\cong k[t]/(t^n-1),

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

k[C_2]\cong k[s]/(s^2-1).

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

e_\pm=\tfrac12(\delta_e\pm \delta_s)

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:

z_1=\delta_e,\qquad z_2=\sum_{\text{transpositions }\tau}\delta_\tau,\qquad z_3=\sum_{\text{3-cycles }\gamma}\delta_\gamma.

Over an algebraically closed field, these class sums act as scalars in any finite-dimensional irreducible representation (compare Schur’s lemma).

Example 1: Cyclic groups CnC_nCn​

Example 2: The order-2 group C2={e,s}C_2=\{e,s\}C2​={e,s}

Example 3: S3S_3S3​ and class sums

Example 1: Cyclic groups $C_n$

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

Example 3: $S_3$ and class sums