Some Representations of the General Linear Group
Sorry for the delays, but it’s the last week of class and everyone came back from the break in a panic.
Okay, let’s look at some examples of group representations. Specifically, let’s take a vector space 
and consider its general linear group 
.
This group comes equipped with a representation already, on the vector space itself! Just use the identity homomorphism 
We often call this the “standard” or “defining” representation. In fact, it’s easy to forget that it’s a representation at all. But it is.
As with any other group, we have dual representations. That is, we immediately get an action of on 
. And we’ve seen it already! When we talked about the coevaluation on vector spaces we worked out how a change of basis affects linear functionals. What we found is that if 
is our action on , then the action on is by the transpose — the dual — of 
. And this is exactly the dual representation.
Also, as with any other group, we have tensor representations — actions on the tensor power 
for any number 
of factors of . How does this work? Well, every vector in 
is a linear combination of vectors of the form 
, where each 
. And we know how to act on these: just act on each tensorand separately. That is,
=\left[\rho(g)\right]\left(v_1\right)\otimes...\otimes\left[\rho(g)\right]\left(v_n\right)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Crho%28g%29%5Cright%5D%28v_1%5Cotimes...%5Cotimes+v_n%29%3D%5Cleft%5B%5Crho%28g%29%5Cright%5D%5Cleft%28v_1%5Cright%29%5Cotimes...%5Cotimes%5Cleft%5B%5Crho%28g%29%5Cright%5D%5Cleft%28v_n%5Cright%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\rho(g)\right](v_1\otimes...\otimes v_n)=\left[\rho(g)\right]\left(v_1\right)\otimes...\otimes\left[\rho(g)\right]\left(v_n\right)")
Then we just extend this action by linearity to all of 
.
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December 2, 2008 -
Posted by John Armstrong |
Algebra, Linear Algebra, Representation Theory
Out of curiosity: where are you heading with this? Were you planning to discuss Schur-Weyl duality?
Actually, I’m going to mention it tomorrow. But I’m not going to actually prove it, or use it as such. What I am going to do is introduce a particular representation that many readers may already be familiar with, but without picking a basis!
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