Kernels and Images of Intertwiners
The next obvious things to consider are the kernel and the image of an intertwining map. So let’s say we’ve got a representation , a representation
, and an intertwiner
defined by the linear map
which satisfies
for all
.
Now the linear map immediately gives us two subspaces: the kernel
and the image
. And it turns out that each of these is actually a subrepresentation. Showing this isn’t difficult. A subrepresentation is just a subspace that gets sent to itself under the action on the whole space, so we just have to check that
always sends vectors in
back to this subspace, and that
always sends vectors in
back into this subspace.
First off, is in the kernel of
if
. Then we calculate
which shoes that is also in the kernel of
.
On the other hand, if is in the image of
, then there is some
so that
. We calculate
And so is also in the image of
.
So we’ve seen that the image and kernel of an intertwining map have well-defined actions of , and so we have subrepresentations. Immediately we can conclude that the coimage
and the cokernel
are quotient representations.