We move on to more theoretical considerations. Let
Let
Either
If
for some scalar
More generally,
Part (1) follows directly from Lemma 1.24: suppose that
For (2), we can find an eigenvalue
For (3), by part (1) we have
As a corollary we obtain
Let
Let
Hence by Schur’s Lemma,
for all
But now, any non-zero
It is possible to give an alternative proof of this using the fact from linear algebra that any commuting set of linear maps from a finite dimensional vector space to itself has a simultaneous eigenvector.
In Schur’s Lemma and Theorem 1.33, we didn’t assume
Any irreducible representation of a finite group is finite-dimensional.
Let
A homomorphism
is called the character group, or dual group, of
Let
Indeed, pick a generator
determines a group isomorphism
In fact, if
For arbitrary groups
Let
be the center of
for all
We call
Homework; mimic the proof that irreducible representations of abelian groups are one-dimensional. ∎
Finally, we can use our classification of the irreducible representations of abelian groups to get a bound on the dimension of the irreducible representations of any finite group.
Let
Restrict the representation to
for all
But this implies that
so has dimension at
most
The group