The other day at tea Theo asked Anton and I a question about the following situation: suppose that A is an extension of X and Y in some symmetric tensor category, what can you say about the tensor square of A? Although the eventual answer seemed to be “you can’t say much,” we ended up working out an interesting example that I’ll share below. I’m hoping one of you can explain to me what’s going on here.
What questions can one ask about a non-semisimple tensor category? First you’d like to know all the irreducible representations. However, since the category is non-semisimple, you still need to understand extensions between these irreducibles. Typically this will split up your category into blocks such that all nontrivial extensions are within a given block, and there are no extensions between different blocks. Then you’d like to understand the indecomposables within each block. Finally you’d like to understand tensor products between these indecomposables. Tensor product will typically mix up the different blocks. In this post I’ll answer most of the above questions for the category of finite dimensional representations of (see this old post for the same questions about a different category).
Such a representation is given by a single matrix, and isomorphism of representations corresponds to conjugation. Irreducible representations correspond to eigenspaces. So the irreducible representations are 1-dimensional spaces that we’ll call where is the eigenvalue. There are no extensions between eigenspaces for different eigenvalues, so the blocks correspond to eigenvalues. Finally, by Jordan normal form, the indecomposable representations are given by an eigenvalue and a Jordan block of size n+1 (we’ll let n denote the number of 1’s off the diagonal, so that ).
What happens under tensor product? The eigenvalues multiply. So the only blocks that have good behavior are the block and the block. We’ll concentrate on the former (though the latter is probably interesting too). What is ? For example, is given by the following matrix:
A simple calculation shows that this matrix is conjugate to one with a Jordan block of size 3 and a Jordan block of size 1. Hence, . A few more calculations like the above suggest (unless I’ve made an error somewhere) that
But this is exactly the tensor product rule for irreducible representations of !? What’s going on here? I suspect this is something I should have already known somehow, but I can’t quite put my finger on it. Also what about the other blocks? How does the tensor product between look? Certainly it will lie inside the block, but which will occur?