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A quick lemma on group representations
*September 23, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: faithful representation, regular representation, symmetric group, tensor powers

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Tags: faithful representation, regular representation, symmetric group, tensor powers

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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 is one where different elements of induce different linear transformations, i.e. is injective. The result is

Lemma 1If is a faithful representation of , then every simple representation of occurs as a direct summand in some tensor power .(more…)