Let 
be the category of finite-dimensional unitary representations of a finite group 
As we discussed last week, is both “bigger” and “smaller” than the category 
of finite-dimensional Hilbert spaces. What this means is that the same Hilbert space 
may appear in as two distinct representations 
and 
, but at the same time hom-spaces in may be smaller because, unlike 
, linear maps in 
must intertwine the actions 
and 
of two unitary representations and 
. In fact, we have already made this “smaller” feature quantitative by deriving the dimension formula
In this lecture we will compare and structurally as opposed to numerically.
One of the simplest things we can say about a linear map 
is that its kernel and image are subspaces of and 
, respectively. It is therefore natural to hope that a 
-equivariant linear map 
would have kernel and image that are “subrepresentations” of and , respectively. To make sense of this we first need to define subobjects in 
Definition 22.1. Let be a unitary representation of 
and let be a vector subspace of 
We say that is -stable if 
for all 
and 
Problem 22.1. Prove that if then 
is a -stable subspace of and 
is a -stable subspace of 
The only reason that the above is not our definition of “subrepresentation” is that we require representations to be nonzero.
Definition 22.2. A subrepresentation of is a nonzero -stable subspace 
In particular, 
is a unitary representation of in its own right provided we restrict each 
to
Now let us recall the main structure theorem we have for linear maps between finite-dimensional vector spaces, which was the main result of Math 202A.
Theorem (SVD): For any linear map between finite-dimensional Hilbert spaces, there exist distinct positive numbers 
and nonzero pairwise orthogonal subspaces 
and 
of and such that
and the restriction of 
to 
has the form 
with 
an isometric isomorphism.
It is an under-appreciated fact that the SVD holds in in a very natural way.
Theorem 22.3 (-equivariant SVD). Let and be unitary representations of and let be a -equivariant linear map. Then, the left singular spaces of are subrepresentations of , the right singular spaces are subrepresentations of 
and the restriction of to is an isomorphism in 
Moreover, the map 
is an isometric isomorphism in 
In 
a space whose only subspace is is necessarily one-dimensional. In 
the situation is not as simple.
Definition 22.1. A unitary representation of is said to be irreducible if the only subrepresentation it contains is itself.
Let 
be the full subcategory of whose objects are irreducible representations. The hom-spaces in this category are highly restricted.
Theorem 22.4 (Schur’s Lemma). If and are irreducible unitary representations of , then any nonzero map is an isomorphism.
Problem 22.2. Prove Schur’s Lemma (it follows directly from the $latex $G-equivariant SVD).$
Another way to state Schur’s Lemma is that if and are irreducible representations which are not isomorphic, then
Together with our dimension formula for 
this yields the following.
Corollary 22.5. The characters of two non-isomorphic irreducible unitary representations and are orthogonal: we have 
Problem 22.3. Prove that the character of any unitary representation of is a class function on Prove that the characters of any two isomorphic representations are equal. Combining these facts, deduce an upper bound on the number of isomorphism classes of irreducible unitary representations of
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