This is a standard result that any beginner in the study of the representation theory would be aware of. I merely present restate the theorem a little(essentially I don’t) and prove it.
Theorem:
Let
be representations of a group
. If
is a
map, then the following are true assuming that
and
are not isomorphic:
1)If both
are irreducible, then
is either an isomorphism or the zero map.
2)If only
is irreducible, then
is either injective or the zero map.
3)If only
is irreducible, then
is either surjective or the zero map
If
are isomorphic as representations, then
is a scalar multiplication map.
Proof:
Let be the respective representations. Here,
are vector spaces over an algebraically closed field.
We tackle the first part of the theorem.
Consider the kernel of , say
, which is set of all
such that
. Since
is a G-linear map,
implies that if
, then
. Hence,
is stable under the action of
which makes it a sub-representation. Since
is irreducible, this means that either
must be zero or
must be the zero map. This proves that
is injective.