Submodules
Fancy words: a submodule is a subobject in the category of 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
-morphism is injective if and only if it’s injective as a linear map; its kernel is zero. We call
the “inclusion map” of the submodule.
In practice, we often identify a -submodule with the image of its inclusion map. We know from general principles that since
is injective, then
is isomorphic to its image, so this isn’t really a big difference. What we can tell, though, is that the action of
sends the image back into itself.
That is, let’s say that is the image of some vector
. I say that for any group element
, acting by
on
gives us some other vector that’s also in the image of
. Indeed, we check that
which is again in the image of , as asserted. We say that the image of
is “
-invariant”.
The flip side of this is that any time we find such a -invariant subspace of
, it gives us a submodule. That is, if
is a
-module, and
is a
-invariant subspace, then we can define a new representation on
by restriction:
. The inclusion map that takes any vector
and considers it as a vector in
clearly intertwines the original action
and the restricted action
, and its kernel is trivial. Thus
constitutes a
-submodule.
As an example, let be any finite group, and let
be its group algebra, which carries the left regular representation
. Now, consider the subspace
spanned by the vector
That is, consists of all vectors for which all the coefficients
are equal. I say that this subspace
is
-invariant. Indeed, we calculate
But this last sum runs through all the elements of , just in a different order. That is,
, and so
carries the one-dimensional trivial representation of
. That is, we’ve found a copy of the trivial representation of
as a submodule of the left regular representation.
As another example, let be one of the symmetric groups. Again, let
carry the left regular representation, but now let
be the one-dimensional space spanned by
It’s a straightforward exercise to show that is a one-dimensional submodule carrying a copy of the signum representation.
Every -module
contains two obvious submodules: the zero subspace
and the entire space
itself are both clearly
-invariant. We call these submodules “trivial”, and all others “nontrivial”.