# The Unapologetic Mathematician

## Morphisms Between Representations

Since every representation of $G$ is a $G$-module, we have an obvious notion of a morphism between them. But let’s be explicit about it.

A $G$-morphism from a $G$-module $(V,\rho_V)$ to another $G$-module $(W,\rho_W)$ is a linear map $T:V\to W$ between the vector spaces $V$ and $W$ that commutes with the actions of $G$. That is, for every $g\in G$ we have $\rho_W(g)\circ T=T\circ\rho_V(g)$. Even more explicitly, if $g\in G$ and $v\in V$, then

$\displaystyle\left[\rho_W(g)\right]\left(T(v)\right)=T\left(\left[\rho_V(g)\right](v)\right)$

We can also express this with a commutative diagram:

For each group element $g\in G$ our representations give us vertical arrows $\rho_V(g):V\to V$ and $\rho_W(g):W\to W$. The linear map $T$ provides horizontal arrows $T:V\to W$. To say that the diagram “commutes” means that if we compose the arrows along the top and right to get a linear map from $V$ to $W$, and if we compose the arrows along the left and bottom to get another, we’ll find that we actually get the same function. In other words, if we start with a vector $v\in V$ in the upper-left and move it by the arrows around either side of the square to get to a vector in $W$, we’ll get the same result on each side. We get one of these diagrams — one of these equations — for each $g\in G$, and they must all commute for $T$ to be a $G$-morphism.

Another common word that comes up in these contexts is “intertwine”, as in saying that the map $T$ “intertwines” the representations $\rho_V$ and $\rho_W$, or that it is an “intertwinor” for the representations. This language goes back towards the viewpoint that takes the representing functions $\rho_V$ and $\rho_W$ to be fundamental, while $G$-morphism tends to be more associated with the viewpoint emphasizing the representing spaces $V$ and $W$.

If, as will usually be the case for the time being, we have a presentation of our group by generators and relations, then we’ll only need to check that $T$ intertwines the actions of the generators. Indeed, if $T$ intertwines the actions of $g$ and $h$, then it intertwines the actions of $gh$. We can see this in terms of diagrams by stacking the diagram for $h$ on top of the diagram for $g$. In terms of equations, we check that

$\displaystyle\begin{aligned}\rho_W(gh)\circ T&=\rho_W(g)\circ\rho_W(h)\circ T\\&=\rho_W(g)\circ T\circ\rho_V(h)\\&=T\circ\rho_V(g)\circ\rho_V(h)\\&=T\circ\rho_V(gh)\end{aligned}$

So if we’re given a set of generators and we can write every group element as a finite product of these generators, then as soon as we check that the intertwining equation holds for the generators we know it will hold for all group elements.

There are also deep connections between $G$-morphisms and natural transformations, in the categorical viewpoint. Those who are really interested in that can dig into the archives a bit.

## 15 Comments	»

1. penultimate line of final equation array needs to have a \rho_V(h) on the right side

Comment by | September 21, 2010 | Reply

2. also, the middle term in the right side of the penultimate line should be p_V(g)

Comment by | September 21, 2010 | Reply

3. Thanks; fixed the typos.

Comment by | September 21, 2010 | Reply

4. [...] group representations. What this means is that if and are -modules, and if we have an injective morphism of modules , then we say that is a “submodule” of . And, just to be clear, a [...]

Pingback by | September 22, 2010 | Reply

5. [...] quick one today. Let’s take two -modules and . We’ll write for the vector space of intertwinors from to . This is pretty appropriate because these are the morphisms in the category of -modules. [...]

Pingback by | September 29, 2010 | Reply

6. [...] Now that we know that images and kernels of -morphisms between -modules are -modules as well, we can bring in a very general [...]

Pingback by | September 30, 2010 | Reply

7. [...] and Commutant Algebras We will find it useful in our study of -modules to study not only the morphisms between them, but the structures that they [...]

Pingback by | October 1, 2010 | Reply

8. [...] Today I’d like to show that the space of homomorphisms between two -modules is “additive”. That is, it satisfies the [...]

Pingback by | October 11, 2010 | Reply

9. [...] representation . This is not the kernel we’ve talked about recently, which is the kernel of a -morphism. This is the kernel of a group homomorphism. In this context, it’s the collection of group [...]

Pingback by | October 29, 2010 | Reply

10. [...] of to a linear endomorphism of , we actually get a homomorphism that sends each element of to a -module endomorphism of [...]

Pingback by | November 1, 2010 | Reply

11. [...] parallel between left and right representations: we have morphisms between right -modules just like we had between left -modules. I won’t really go into the details — they’re pretty [...]

Pingback by | November 3, 2010 | Reply

12. [...] so must be an intertwinor. And so we conclude [...]

Pingback by | November 12, 2010 | Reply

13. [...] of all, functoriality of restriction is easy. Any intertwinor between -modules is immediately an intertwinor between the restrictions and . Indeed, all it has [...]

Pingback by | December 1, 2010 | Reply

14. [...] let’s start on the left with a linear map that intertwines the action of each subgroup element . We want to extend this to a linear map from to that [...]

Pingback by | December 3, 2010 | Reply

15. [...] To put it another way, is the vector specifies for the point . On the other hand, is the image of the vector specifies for the point . If these two vectors are the same for every , then we say that “intertwines” the two vector fields, or that and are “-related”. The latter term is a bit awkward, which is why I prefer the former, especially since it does have that same commutative-diagram feel as intertwinors between representations. [...]

Pingback by | June 3, 2011 | Reply

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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”).

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