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 ![\left[\sigma(a)\right]\left(f(v)\right)=f\left(\left[\rho(a)\right](v)\right)](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%3Df%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29&bg=e6e6e6&fg=333333&s=0 "\left[\sigma(a)\right]\left(f(v)\right)=f\left(\left[\rho(a)\right](v)\right)")
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
\right)=\left[\sigma(a)\right]\left(f(v)\right)\\=\left[\sigma(a)\right](0)=0\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Df%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%5C%5C%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%280%29%3D0%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}f\left(\left[\rho(a)\right](v)\right)=\left[\sigma(a)\right]\left(f(v)\right)\\=\left[\sigma(a)\right](0)=0\end{aligned}")
which shoes that ](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29&bg=e6e6e6&fg=333333&s=0 "\left[\rho(a)\right](v)")
is also in the kernel of .
On the other hand, if 
is in the image of , then there is some so that 
. We calculate
=\left[\sigma(a)\right]\left(f(v)\right)\\=f\left(\left[\rho(a)\right](v)\right)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%28w%29%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%5C%5C%3Df%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[\sigma(a)\right](w)=\left[\sigma(a)\right]\left(f(v)\right)\\=f\left(\left[\rho(a)\right](v)\right)\end{aligned}")
And so ](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Csigma%28a%29%5Cright%5D%28w%29&bg=e6e6e6&fg=333333&s=0 "\left[\sigma(a)\right](w)")
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.
December 12, 2008
Posted by John Armstrong |
Algebra, Representation Theory |
2 Comments
