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
.
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
.
June 8, 2010 at 8:11 pm
[…] the regular representation is faithful, it follows (by a lemma of Burnside), that every representation is a direct factor of some direct sum of tensor powers . These […]