With the group algebra in hand, we now define a “-module” to be a module for the group algebra of . That is, it’s a (finite-dimensional) vector space and a bilinear map . This map must satisfy and .
This is really the same thing as a representation, since we may as well pick a basis for and write . Then for any we can write
That is, is a linear map from to itself, with its matrix entries given by . We define this matrix to be , which must be a representation because of the conditions on above.
Conversely, if we have a matrix representation , we can define a module map for as
where we apply the matrix to the column vector . This must satisfy the above conditions, since they reflect the fact that is a representation.
In fact, to define , all we really need to do is to define it for the basis elements . Then linearity will take care of the rest of the group algebra. That is, we can just as well say that a -module is a vector space and a function satisfying the following three conditions:
- is linear in : .
- preserves the identity: .
- preserves the group operation: .
The difference between the representation viewpoint and the -module viewpoint is that representations emphasize the group elements and their actions, while -modules emphasize the representing space . This viewpoint will be extremely helpful when we want to consider a representation as a thing in and of itself. It’s easier to do this when we think of it as a vector space equipped with the extra structure of a -action.