The Unapologetic Mathematician

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 $V$ and consider its general linear group $\mathrm{GL}(V)$.

This group comes equipped with a representation already, on the vector space $V$ itself! Just use the identity homomorphism $\mathrm{GL}(V)\rightarrow\mathrm{GL}(V)$ 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 $\mathrm{GL}(V)$ on $V^*$. 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 $\rho(g)$ is our action on $V$, then the action on $V^*$ is by the transpose — the dual — of $\rho(g)^{-1}=\rho\left(g^{-1}\right)$. And this is exactly the dual representation.

Also, as with any other group, we have tensor representations — actions on the tensor power $V^{\otimes n}=V\otimes...\otimes V$ for any number $n$ of factors of $V$. How does this work? Well, every vector in $V\otimes...\otimes V$ is a linear combination of vectors of the form $v_1\otimes...\otimes v_n$, where each $v_k\in V$. And we know how to act on these: just act on each tensorand separately. That is,

$\displaystyle\left[\rho(g)\right](v_1\otimes...\otimes v_n)=\left[\rho(g)\right]\left(v_1\right)\otimes...\otimes\left[\rho(g)\right]\left(v_n\right)$

Then we just extend this action by linearity to all of $V^{\otimes n}$.

December 2, 2008