Maschke corollary (regular representation decomposition)

Let $G$ be a finite group and $k$ an algebraically closed field such that $\operatorname{char}(k)\nmid |G|$ . Let $\{V_1,\dots,V_r\}$ be representatives of the isomorphism classes of finite-dimensional irreducible representations of $G$ , and set $d_i=\dim_k V_i$ . Then the regular representation decomposes as a left $k[G]$ -module:

k[G]\;\cong\;\bigoplus_{i=1}^r d_i\,V_i.

Thus each $V_i$ occurs with multiplicity $d_i$ .

Explanation

By Maschke's theorem, the regular representation is completely reducible. Algebraic closedness identifies the multiplicity of $V_i$ with $\dim_k V_i$ .

Remarks

A closely related semisimple-algebra statement is that the group algebra admits a Wedderburn decomposition

k[G]\;\cong\;\bigoplus_{i=1}^r \operatorname{End}_k(V_i)\;\cong\;\bigoplus_{i=1}^r \mathrm{Mat}_{d_i}(k),

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:

|G|=\dim_k(k[G])=\sum_i d_i^2.

Examples

$S_3$ (order $6$ ). Over $\mathbb C$ , the irreducibles are $\mathbf{1}$ (trivial), $\mathrm{sgn}$ (sign), and the $2$ -dimensional standard representation $V$ . The corollary gives $\mathbb C[S_3]\;\cong\; 1\cdot \mathbf{1}\;\oplus\;1\cdot \mathrm{sgn}\;\oplus\;2\cdot V.$ The dimensions sum to $1+1+2\cdot 2=6$ .

$C_n$ (order $n$ ). All irreducibles are $1$ -dimensional characters $\chi_0,\dots,\chi_{n-1}$ , hence $\mathbb C[C_n]\;\cong\;\chi_0\oplus\chi_1\oplus\cdots\oplus\chi_{n-1}.$

$D_8$ (order $8$ ). The group $D_8$ 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 $4\cdot 1 + 2\cdot 2 = 8$ .