Decomposability
Today I’d like to cover a stronger condition than reducibility: decomposability. We say that a module is “decomposable” if we can write it as the direct sum of two nontrivial submodules
and
. The direct sum gives us inclusion morphisms from
and
into
, and so any decomposable module is reducible.
What does this look like in terms of matrices? Well, saying that means that we can write any vector
uniquely as a sum
with
and
. Then if we have a basis
of
and a basis
of
, then we can write
and
uniquely in terms of these basis vectors. Thus we can write any vector
uniquely in terms of the
, and so these constitute a basis of
.
If we write the matrices in terms of this basis, we find that the image of any
can be written in terms of the others because
is
-invariant. Similarly, the
-invariance of
tells us that the image of each
can be written in terms of the others. The same reasoning as last time now allows us to conclude that the matrices of the
all have the form
Conversely, if we can write each of the in this form, then this gives us a decomposition of
as the direct sum of two
-invariant subspaces, and the representation is decomposable.
Now, I said above that decomposability is stronger than reducibility. Indeed, in general there do exist modules which are reducible, but not decomposable. Indeed, in categorical terms this is the statement that for some groups there are short exact sequences which do not split. To chase this down a little further, our work yesterday showed that even in the reducible case we have the equation
. This
is the representation of
on the quotient space, which gives our short exact sequence
But in general this sequence may not split; we may not be able to write as
-modules. Indeed, we’ve seen that the representation of the group of integers
is indecomposable.