The Unapologetic Mathematician

Representing Product Groups

An important construction for groups is their direct product. Given two groups $G$ and $H$ we take the cartesian product of their underlying sets $G\times H$ and put a group structure on it by multiplying component-by-component. That is, the product of $(g_1,h_1)$ and $(g_2,h_2)$ is $(g_1g_2,h_1h_2)$. Representations of product groups aren’t really any different from those of any other group, but we have certain ways of viewing them that will come in handy.

The thing to notice is that we have copies of both $G$ and $H$ inside $G\times H$. Indeed, we can embed $G$ into $G\times H$ by sending $g$ to $(g,e_H)$, which clearly preserves the multiplication. Similarly, the map $h\mapsto(e_G,h)$ embeds $H$ into $G$. The essential thing here is that the transformations coming from $G$ and those coming from $H$ commute with each other. Indeed, we calculate

\displaystyle\begin{aligned}(g,e_H)(e_G,h)&=(ge_G,e_Hh)\\&=(g,h)\\&=(e_Gg,he_H)\\&=(e_G,h)(g,e_H)\end{aligned}

Also, every transformation in $G\times H$ is the product of one from $G$ and one from $H$.

The upshot is that a representation of $G\times H$ on a space $V$ provides us with a representation of $G$, and also one of $H$ on the space $V$. Further, transformation in the representation of $G$ must commute with every transformation in the representation of $V$. Conversely, if we have a representation of each factor group on the same space $V$, then we have a representation of the product group, but only if all the transformations in each representation commute with all the transformations in the other.

So what can we do with this? Well, it turns out that it’s pretty common to have two separate group actions on the same module, and to have these two actions commute with each other like this. Whenever this happens we can think of it as a representation of the product group, or as two commuting representations.

In fact, there’s another way of looking at it: remember that a representation of a group $G$ on a space $V$ can be regarded as a module for the group algebra $\mathbb{C}[G]$. If we then add a commuting representation of a group $H$, we can actually regard it as a representation on the $G$-module instead of just the underlying vector space. That is, instead of just having a homomorphism $H\to\mathrm{End}(V)$ that sends each element of $H$ to a linear endomorphism of $V$, we actually get a homomorphism $H\to\mathrm{End}_G(V)$ that sends each element of $H$ to a $G$-module endomorphism of $V$.

Indeed, let’s write our action of $G$ with the group homomorphism $\rho$ and our action of $H$ with the group homomorphism $\sigma$. Now, I’m asserting that each $\sigma(h)$ is an intertwinor for the action of $G$. This means that for each $g\in G$, it satisfies the equation $\rho(g)\sigma(h)=\sigma(h)\rho(g)$. But this is exactly what it means for the two representations to commute!

Some notation will be helpful here. If the vector space $V$ carries a representation of a group $G$, we can hang a little tag on it to remind ourselves of this, writing ${}_GV$. That is, ${}_GV$ is a $G$-module, and not just a vector space. If we now add a new representation of a group $H$ that commutes with the original representation, we just add a new tag: ${}_{HG}V$. Of course, the order of the tags doesn’t really matter, so we could just as well write ${}_{GH}V$. Either way, this means that we have a representation of $G\times H$ on $V$.

November 1, 2010 -