Semisimplicity and Schur’s Lemma
A representation 
of a finite group is said to be irreducible if there is no nontrivial vector subspace 
of 
which is left stable under the action of 
.
If there is an inner-product lurking in the background, and 
is a unitary representation, then for every 
there exists a unique 
such that 
. If is such that:
Then it follows that is such that:
But since the representation is unitary, 
, thus every unitary representation of decomposes into an orthogonal direct sum of irreducible ones. This is the definition of semisimplicity.
Now fix some representation . If is irreducible, then the only maps 
with the property that for all 
the following diagram commutes:
are the scalar multiples of the identity.
To see this not that if 
is an eigenvalue of then the eigenspace of corresponding to is stable under all of and must thus be the whole space. Over 
every linear operator has at least one eigenvalue.
More generally, suppose that decomposes as:
with each of
and irreducible. If the representation of restricted to is not isomorphic to the representation of on , then the the only 
which commutes with the action of are those of the form:
where
denotes the identity on and 
denotes the identity of 
.
If they are isomorphic, then we can write:
where
is a multiplicity space of dimension 
. In this basis the representation looks like:
where
is the restriction of to and 
is the identity matrix on .
The which commute with the action of are now of the form:
where
denotes the identity matrix acting on , and 
is any matrix acting on .
Finally if 
is an arbitrary representation, and the underlying vector space decomposes as
with the
denoting pairwise non-isomorphic irreducible representations, and the 
their multiplicity spaces, then the representation looks like:
and the
which commute with the action of are those of the form:
This is more or less Schur’s lemma.