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…)