## A quick lemma on group representationsSeptember 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: , , ,

So, since I’ll be talking about the symmetric group a bit, and since I still don’t have enough time for a deep post on it, I’ll take the opportunity to cover a quick and relevant lemma in group representation theory (referring as usual to the past blog post as background).

A faithful representation of a finite group ${G}$ is one where different elements of ${G}$ induce different linear transformations, i.e. ${G \rightarrow Aut(V)}$ is injective. The result is

Lemma 1 If ${V}$ is a faithful representation of ${G}$, then every simple representation of ${G}$ occurs as a direct summand in some tensor power ${V^{\otimes p}}$  (more…)