Knowlpedia

Group algebra

The associative algebra k[G] whose basis is a group G and whose multiplication extends the group law bilinearly.

Let GG be a finite group and kk a field. The group algebra k[G]k[G] (also written kGkG) is the kk- with basis { δg:g∈G }\{\,\delta_g : g\in G\,\} and multiplication determined by

δg⋅δh=δgh(g,h∈G),\delta_g\cdot \delta_h = \delta_{gh}\quad (g,h\in G),

extended kk-bilinearly. Thus every element has a unique expression

x=∑g∈Gag δg(ag∈k),x=\sum_{g\in G} a_g\,\delta_g\qquad (a_g\in k),

and multiplication is

(∑gagδg)(∑hbhδh)=∑g,hagbh δgh.\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 element of k[G]k[G] is δe\delta_e, where ee is the identity of GG.

Representations as modules

A (finite-dimensional) ρ:G→GL(V)\rho:G\to \mathrm{GL}(V) on a kk-vector space VV extends uniquely to a kk-algebra homomorphism

ρ~:k[G]→Endk(V),ρ~ ⁣(∑gagδg)=∑gag ρ(g).\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 GG is the same as giving a left k[G]k[G]-module structure on VV (i.e. an action k[G]×V→Vk[G]\times V\to V that is kk-bilinear and associative). In this correspondence:

Examples

Example 1: Cyclic groups CnC_n

Let G=Cn=⟨t∣tn=e⟩G=C_n=\langle t\mid t^n=e\rangle. Then

k[Cn]≅k[t]/(tn−1),k[C_n]\cong k[t]/(t^n-1),

via δtm↦tm\delta_{t^m}\mapsto t^m. This realizes k[Cn]k[C_n] as a commutative kk-algebra.

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

Here k[C2]=kδe⊕kδsk[C_2]=k\delta_e\oplus k\delta_s with δs2=δe\delta_s^2=\delta_e. So

k[C2]≅k[s]/(s2−1).k[C_2]\cong k[s]/(s^2-1).

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

e±=12(δe±δs)e_\pm=\tfrac12(\delta_e\pm \delta_s)

satisfy e±2=e±e_\pm^2=e_\pm and e+e−=0e_+e_-=0, giving a decomposition k[C2]≅k×kk[C_2]\cong k\times k. (This is a concrete instance of semisimplicity in characteristic not dividing ∣G∣|G|.)

Example 3: S3S_3 and class sums

For G=S3G=S_3, k[S3]k[S_3] is 66-dimensional with basis {δσ:σ∈S3}\{\delta_\sigma:\sigma\in S_3\}. The center Z(k[S3])Z(k[S_3]) is spanned by sums over :

z1=δe,z2=∑transpositions τδτ,z3=∑3-cycles γδγ.z_1=\delta_e,\qquad z_2=\sum_{\text{transpositions }\tau}\delta_\tau,\qquad z_3=\sum_{\text{3-cycles }\gamma}\delta_\gamma.

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