What is the regular representation of a finite group \(G\)? State its character, decomposition into irreducible representations, and connection to the group algebra \(\mathbb{C}[G]\).
Solution
The left regular representation of a finite group \(G\) is the action on its group algebra \[ \mathbb C[G] \] by left multiplication: \[ g\cdot e_h=e_{gh}. \] It has dimension \(|G|\) and is faithful.
Its character is \[ \chi_{\mathrm{reg}}(g)= \begin{cases} |G|,&g=1,\\ 0,&g\ne1. \end{cases} \] Indeed, for \(g\ne1\), left multiplication fixes no basis vector \(e_h\).
Let \(V_1,\dots,V_r\) be the irreducible complex representations, with characters \(\chi_i\) and dimensions \(d_i\). The multiplicity of \(V_i\) in the regular representation is \[ \langle\chi_{\mathrm{reg}},\chi_i\rangle =\frac1{|G|}|G|\chi_i(1)=d_i. \] Hence \[ \mathbb C[G]\cong\bigoplus_i V_i^{\oplus d_i} \] as a left \(G\)-module, and therefore \[ |G|=\sum_i d_i^2. \] As a semisimple algebra, Artin–Wedderburn gives \[ \mathbb C[G]\cong\prod_i M_{d_i}(\mathbb C). \]