# The Unapologetic Mathematician

## Group Representations

We’ve now got the general linear group $\mathrm{GL}(V)$ of all invertible linear maps from a vector space $V$ to itself. Incidentally this lives inside the endomorphism algebra $\hom_\mathbf{Vect}(V,V)$ of all linear transformations from $V$ to itself. In fact, in ring-theory terms it’s the group of units of that algebra. So what can we do with it?

One of the biggest uses is to provide representations for other algebraic structures. Let’s say we’ve got some abstract group. It’s a set with some binary operation defined on it, sure, but what does it do? We’ve seen groups acting on sets before, where we interpret a group element as a permutation of an actual collection of elements. Alternatively, an action of a group $G$ is a homomorphism from $G$ to the group of permutations of some set $S$$\hom_\mathbf{Set}(S,S)$.

Another concrete representation of a group is as symmetries of some vector space. That is, we’re interested in homomorphisms $\rho:G\rightarrow\mathrm{GL}(V)$. A “representation” of a group $G$ is a vector space $V$ with such a homomorphism.

In fact, this extends the notion of a group acting on a set. Indeed, for any set $S$ we can build the free vector space $\mathbb{F}[S]$ with a basis vector $e_s$ for each $s\in S$. Given a permutation $\pi$ on $S$ we get a linear map $\mathbb{F}[\pi]:\mathbb{F}[S]\rightarrow\mathbb{F}[S]$ defined by setting $\mathbb{F}[\pi](e_s)=e_{\pi(s)}$ and extending by linearity.

We thus get a homomorphism from the group of permutations of $S$ to $\mathrm{GL}(\mathbb{F}[S])$. And then if we have a group action on $S$ we can promote it to a representation on the vector space $\mathbb{F}[S]$. We call such a representation a “permutation representation”.

October 23, 2008