Maschke corollary (regular representation decomposition)

Let $G$ be a finite group and let $k$ be a field with $\operatorname{char}(k)\nmid |G|$ (e.g. $k=\mathbb C$). By Maschke's theorem , every finite-dimensional representation of $G$ over $k$ is completely reducible ; equivalently every $k[G]$-module is semisimple .

Assume moreover that $k$ is algebraically closed, and let $\{V_1,\dots,V_r\}$ be a complete set of pairwise non-isomorphic finite-dimensional irreducible representations of $G$, with $d_i=\dim_k(V_i)$.

Corollary (regular representation decomposition)

As a left $kG$-module (i.e. as a $G$-representation), the regular representation decomposes as

In particular, each irreducible $V_i$ occurs inside $k[G]$ with multiplicity exactly $\dim(V_i)$.

A closely related (often packaged together) semisimple-algebra statement is that the group algebra admits a Wedderburn decomposition

where the second isomorphism uses $\dim(V_i)=d_i$ and algebraic closedness of $k$.

Taking dimensions in the module decomposition gives the formula sum of squares of degrees :

Examples

1. $S_3$ (order $6$).
   Over $\mathbb C$, the irreducibles are $\mathbf{1}$ (trivial), $\mathrm{sgn}$ (sign), and the $2$-dimensional standard $V$. The corollary gives

   $$\mathbb C[S_3]\;\cong\; 1\cdot \mathbf{1}\;\oplus\;1\cdot \mathrm{sgn}\;\oplus\;2\cdot V.$$

   Dimension check: $1+1+2\cdot 2=6$.

2. $C_n$ (order $n$).
   All irreducibles are $1$-dimensional characters $\chi_0,\dots,\chi_{n-1}$, hence

   $$\mathbb C[C_n]\;\cong\;\chi_0\oplus\chi_1\oplus\cdots\oplus\chi_{n-1}.$$

3. $D_8$ (order $8$).
   $D_8$ has four $1$-dimensional irreducibles $\chi_1,\dots,\chi_4$ and one $2$-dimensional irreducible $V$, so

   $$\mathbb C[D_8]\;\cong\;\chi_1\oplus\chi_2\oplus\chi_3\oplus\chi_4\oplus 2\cdot V,$$

   and $4\cdot 1 + 2\cdot 2 = 8$.