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.

September 21, 2010 - Posted by John Armstrong | Algebra, Group theory, Representation Theory

15 Comments »

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

Comment by emilelahner | September 21, 2010 | Reply
also, the middle term in the right side of the penultimate line should be p_V(g)

Comment by emilelahner | September 21, 2010 | Reply
Thanks; fixed the typos.

Comment by John Armstrong | September 21, 2010 | Reply
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