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