A quick lemma on group representations September 23, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: faithful representation, regular representation, symmetric group, tensor powers
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 1 If is a faithful representation of , then every simple representation of occurs as a direct summand in some tensor power .
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 .