Modules
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.