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.
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,
Then we just extend this action by linearity to all of .