# 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”.

## 13 Comments	»

1. [...] We’ve defined a representation of the group as a homomorphism for some vector space . But where did we really use the fact that [...]

Pingback by | October 24, 2008 | Reply

2. [...] We’ve seen how group representations are special kinds of algebra representations. But even more general than that is the representation [...]

Pingback by | October 27, 2008 | Reply

3. [...] Now let’s narrow back in to representations of algebras, and the special case of representations of groups, but with an eye to the categorical interpretation. So, representations are functors. And this [...]

Pingback by | October 28, 2008 | Reply

4. [...] general linear group acts on the hom-set by conjugation — basis changes. In fact, this is a representation of the group, but I’m not ready to go into that detail right now. What I can say is that the [...]

Pingback by | October 30, 2008 | Reply

5. [...] let’s look at some examples of group representations. Specifically, let’s take a vector space and consider its general linear group [...]

Pingback by | December 2, 2008 | Reply

6. [...] to the symmetric group . In fact, these are the image of the usual generators of under the permutation representation. They just rearrange the order of the basis [...]

Pingback by | August 26, 2009 | Reply

7. [...] course it was the fall of 2008 before I even defined a group representation in the main exposition, so some of this information has been a long time coming. But still, what [...]

Pingback by | January 11, 2010 | Reply

8. [...] course it was the fall of 2008 before I even defined a group representation in the main exposition, so some of this information has been a long time coming. But still, what [...]

Pingback by | August 28, 2010 | Reply

9. [...] let’s go in a completely different direction! I want to talk about the representation theory of permutation groups. Now at least on the surface you might not think there’s a lot [...]

Pingback by | September 7, 2010 | Reply

10. [...] specifically concerned with complex representations of these groups. That is, we want to pick some complex vector space , and for each permutation we [...]

Pingback by | September 8, 2010 | Reply

11. [...] Sample Representations As promised, we want to see some examples of matrix representations for those who might not have seen much of them before. These are homomorphisms from a group to the [...]

Pingback by | September 13, 2010 | Reply

12. [...] and Descending Representations Let’s recall that a group representation is, among other things, a group homomorphism. This has a few [...]

Pingback by | October 29, 2010 | Reply

13. [...] Lie groups are groups, they have representations — homomorphisms to the general linear group of some vector space or another. But since is a [...]

Pingback by | June 13, 2011 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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