Since every representation of is a -module, we have an obvious notion of a morphism between them. But let’s be explicit about it.
A -morphism from a -module to another -module is a linear map between the vector spaces and that commutes with the actions of . That is, for every we have . Even more explicitly, if and , then
We can also express this with a commutative diagram:
For each group element our representations give us vertical arrows and . The linear map provides horizontal arrows . To say that the diagram “commutes” means that if we compose the arrows along the top and right to get a linear map from to , 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 in the upper-left and move it by the arrows around either side of the square to get to a vector in , we’ll get the same result on each side. We get one of these diagrams — one of these equations — for each , and they must all commute for to be a -morphism.
Another common word that comes up in these contexts is “intertwine”, as in saying that the map “intertwines” the representations and , or that it is an “intertwinor” for the representations. This language goes back towards the viewpoint that takes the representing functions and to be fundamental, while -morphism tends to be more associated with the viewpoint emphasizing the representing spaces and .
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 intertwines the actions of the generators. Indeed, if intertwines the actions of and , then it intertwines the actions of . We can see this in terms of diagrams by stacking the diagram for on top of the diagram for . In terms of equations, we check that
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 -morphisms and natural transformations, in the categorical viewpoint. Those who are really interested in that can dig into the archives a bit.