Maschke's theorem

Let $𝐺$ be a finite group. Then the group ring $𝕂 [𝐺]$ is semisimple iff $c h a r 𝕂$ does not divide $| 𝐺 |$ . #m/thm/rep2

Proof of semisimplicity for coprime characteristic

Let $0 \neq 𝑈 < 𝕂 [𝐺] 𝕂 [𝐺]$ be a nonzero proper (left) submodule. Since $𝕂 [𝐺]$ is a $𝕂$ -vector space, we have $𝑈 < 𝕂 𝕂 [𝐺]$ , and we can find a complement subspace $𝑊 0 = 𝑈 𝑐$ , such that $𝕂 [𝐺] = 𝕂 𝑈 \oplus 𝑊 0$ . We can define the $𝕂$ -linear projection

𝜑 : 𝕂 [𝐺] ↠ 𝑈

with $k e r 𝜑 = 𝑊 0$ . Let

𝜗 : 𝕂 [𝐺] \to 𝕂 [𝐺] 𝑣 \mapsto 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑔 - 1 𝜑 (𝑔 𝑣)

which is $𝕂$ -linear. Note $i m 𝜗 \leq 𝕂 𝑈$ . We claim it is also $𝕂 [𝐺]$ -linear: Indeed, for all $𝑥 \in 𝐺$ , we have

𝜗 (𝑥 𝑣) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑔 - 1 𝜑 (𝑔 𝑥 𝑣) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑥 𝑥 - 1 𝑔 - 1 𝜑 (𝑔 𝑥 𝑣) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑥 (𝑔 𝑥) - 1 𝜑 (𝑔 𝑥 𝑣) = 1 | 𝐺 | \sum ℎ \in 𝐺 𝑥 ℎ - 1 𝜑 (ℎ 𝑣) = 𝑥 | 𝐺 | \sum ℎ \in 𝐺 ℎ - 1 𝜑 (ℎ 𝑣) = 𝑥 𝜗 (𝑣) .

If $𝑢 \in 𝑈$ , then

𝜗 (𝑢) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑔 - 1 𝜑 (𝑔 𝑢) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 𝑢 = 𝑢

so $𝜗$ is a projection operator onto $𝑈$ . Letting $𝑊 = k e r 𝜗$ we have that $𝑊$ is a $𝕂 [𝐺]$ -submodule, hence

𝕂 [𝐺] = 𝕂 [𝐺] 𝑈 \oplus 𝑊 .

By induction, it follows $𝕂 [𝐺]$ is semisimple.

The above proof gives a construction of a complementary submodule for any submodule, which we call Maschke's algorithm.

In terms of unitary irreps

Every unitary representation is the direct sum of unitary irreps, and thus any representation of a compact group is the direct sum of unitary irreps. #m/thm/rep

Proof

Follows from the fact that The orthogonal complement of an invariant subspace under a unitary operator is invariant and Every finite complex representation of a compact group is equivalent to a unitary representation.

This core statement of group representation theory allows for the Decomposition of a representation, and therefore reduces the task of classifying representations to classifying finite ones.

Maschke's theorem

In terms of unitary irreps

See also