Reducibility
We say that a module is “reducible” if it contains a nontrivial submodule. Thus our examples last time show that the left regular representation is always reducible, since it always contains a copy of the trivial representation as a nontrivial submodule. Notice that we have to be careful about what we mean by each use of “trivial” here.
If the -dimensional representation
has a nontrivial
-dimensional submodule
—
and
— then we can pick a basis
of
. And then we know that we can extend this to a basis for all of
:
.
Now since is a
-invariant subspace of
, we find that for any vector
and
the image
is again a vector in
, and can be written out in terms of the
basis vectors. In particular, we find
, and all the coefficients of
through
are zero. That is, the matrix of
has the following form:
where is an
matrix,
is an
matrix, and
is an
matrix. And, in fact, this same form holds for all
. In fact, we can use the rule for block-multiplying matrices to find:
and we see that actually provides us with the matrix for the representation we get when restricting
to the submodule
. This shows us that the converse is also true: if we can find a basis for
so that the matrix
has the above form for every
, then the subspace spanned by the first
basis vectors is
-invariant, and so it gives us a subrepresentation.
As an example, consider the defining representation of
, which is a permutation representation arising from the action of
on the set
. This representation comes with the standard basis
, and it’s easy to see that every permutation leaves the vector
— along with the subspace
that it spans — fixed. Thus
carries a copy of the trivial representation as a submodule of
. We can take the given vector as a basis and throw in two others to get a new basis for
:
.
Now we can take a permutation — say — and calculate its action in terms of the new basis:
The others all work similarly. Then we can write these out as matrices:
Notice that these all have the required form:
Representations that are not reducible — those modules that have no nontrivial submodules — are called “irreducible representations”, or sometimes “irreps” for short. They’re also called “simple” modules, using the general term from category theory for an object with no nontrivial subobjects.