I came across this question in my notes : If $G$ is finite group and $V$ is finite dimensional $CG$ module ( $C$ complex ) , how do i show that there exist a positive definite Hermitian Form $(.,.)$ for $($gv, gw) for all $g$ and $v$,$w$. And show that V is semisimple without using Maschke's theorem.
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The basic idea is that you know that $V$ admits some inner product $\langle -,-\rangle_0$. Then, just normalize it by definiing $$\langle u,v\rangle=\frac{1}{|G|}\sum_{g\in G}\langle gu,gv\rangle_0$$ You can then prove Maschke's theorem which says that $\mathbb{C}[G]$ is semisimple by showing that every $\mathbb{C}[G]$-submodule $W\leqslant V$ has a complement by showing that $W^\perp$ (with respect to $\langle-,-\rangle$) is such a subspace--which of course proves the semisimplicity of $\mathbb{C}[G]$ since $V$ was arbitrary. |
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Another interesting fact is the following. Let $V$ be an arbitrary representation of $G$ (it could be infinite-dimensional). A given $v\in V$ is contained in a finite-dimensional subrepresentation: $$V_v=\mbox{span}\{g\cdot v:g\in G\}$$ (it is finite dimensional since $G$ is a finite group). The answer of Alex Youcis proves that a finite-dimensional representation is semisimple. Hence, it follows that every $v\in V$ is contained in a semisimple subrepresentation, and so we can conclude that $V$ itself is semisimple. Therefore, all representations of a finite group (even infinite-dimensional) is semisimple. |
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