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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 .

To prove this, let be the character of and the character of , for some irreducible in . Then we need to show for some , in view of the orthonormality relations,

Now, let be the set of values assumed by and let be the set where takes the value . If , then (1) implies

for all . But this implies that each by taking a van der Monde determinant. If, say, —by faithfulness iff —then , which implies .

Note that the proof (due to Brauer) actually gives an effective bound: we can take the tensor power to be at most , where is as in the proof of the result. This follows again from van der Monde determinants.

The case that interests us is the symmetric group , where we have a canonical regular representation spanned by basis vectors with for . This is faithful, so we find that every simple representation of is a summand of some .

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