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.