Jordan normal form and non-semisimple representaton theory

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 \mathbb{C}[x] (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 V_\lambda^0 where \lambda 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 V_\lambda^n are given by an eigenvalue \lambda and a Jordan block of size n+1 (we’ll let n denote the number of 1’s off the diagonal, so that dim V_\lambda^n = n+1).

What happens under tensor product? The eigenvalues multiply. So the only blocks that have good behavior are the \lambda = 1 block and the \lambda = 0 block. We’ll concentrate on the former (though the latter is probably interesting too). What is V_1^n \otimes V_1^m? For example, V_1^1 \otimes V_1^1 is given by the following matrix:

\left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1\end{array}\right)

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, V_1^1 \otimes V_1^1 \cong V_1^2 \oplus V_1^0. A few more calculations like the above suggest (unless I’ve made an error somewhere) that

V_1^n \otimes V_1^m \cong \bigoplus_{|m-n|\leq i \leq |m+n| \text{ and } i \equiv m+n \mod 2} V_1^i.

But this is exactly the tensor product rule for irreducible representations of SL_2!? 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 V_\lambda^n \otimes V_\mu^m look? Certainly it will lie inside the \lambda \mu block, but which V_{\lambda \mu}^i will occur?

11 thoughts on “Jordan normal form and non-semisimple representaton theory

  1. anon appears to be typing with a phone. You can find Weil II here. The section in question is labeled “Jacobson-Morozov.”

    There isn’t anything interesting to say about the tensor structure of blocks with different eigenvalues (over an algebraically closed field), since you can shift by a variable substitution, take tensor product, and shift back by the product of eigenvalues. You might as well assume x is nilpotent, and then use the equivalence indicated by anon.

  2. Anon-

    With the category structure Noah is talking about, J-M isn’t an equivalence of categories, just a bijection between irreducible sl_2 reps and *indecomposible* nilpotent representations of the free monoid with one generator. There are many more maps in the latter category.

  3. Oh, yeah. The standard map \mathbb{C}[x]/(x^2) \to \mathbb{C}[x]/(x) doesn’t work on the sl_2 side, and Schur’s lemma also fails. Even so, I think the bijection on indecomposables extends to a bijection on objects that respects tensor products (with nilpotents given by 1 \otimes N_2 + N_1 \otimes 1).

  4. You might be interested to know that Martin Herschend has studied this problem from the point of view of quiver representations, where the algebra C[x] corresponds to the quiver with one vertex and one loop. I’m not sure he discusses the relationship with sl2, though.

  5. I’m a bit confused, probably because I made a mistake somewhere, but when I tried to compute tensor products in the 0-eigenvalue block I did not get the sl_2 tensor product. I only got that when I looked at the 1-eigenvalue block.

  6. That’s probably because I’m using the wrong tensor product. If I take the correct tensor structure on unipotents, the tensor product of (V,1+A) and (W,1+B) for A and B nilpotent is
    (V \otimes W, 1 \otimes 1 + 1 \otimes B + A \otimes 1 + A \otimes B). If I do a variable substutition, changing x to x+1, this changes my spaces to (V,A) and (W,B). The tensor product is shifted to
    (V \otimes W, 1 \otimes B + A \otimes 1 + A \otimes B), which is equivalent under a basis change to
    (V \otimes W, 1 \otimes B + A \otimes 1).
    This is the tensor product on nilpotents I was using, and it is almost never equivalent to (V \otimes W, A \otimes B).

    I think the tensor I used is the product for the additive algebraic group (or its Lie algebra), which as Ben noted, lives inside sl_2. It is somehow dual to the correct one.

  7. In a spirit of helpfulness, I point out that,
    “The other day at tea Theo asked Anton and I a question about the following situation:”
    should read, “The other day at tea, Theo asked Anton and me a question about the following situation:”

    I make this remark knowing that my remarks are often replete with technical and grammatical errors, missing characters, and misspellings. Our ears get confused when we don’t often hear the correct use of the objective case.

  8. Interesting post. The oldest references I know of to this problem are Aitken (1935) and a follow-up by Littlewood (1936) (need subscriptions for those links…). Littlewood uses some theory of symmetric functions to not only describe the tensor product structure of Jordan blocks, but goes so far as to describe the effect of applying any Schur functor to a Jordan block (in terms of generating functions). Fair warning that the vocabulary was quite different in the 30’s though… google helped me sort through it (e.g. you won’t see “tensor” anywhere in the papers).

    The relationship with SL2 is interesting, I have wondered if we might find more connections between tensor products of quiver representations and other more classically studied tensor products. Or maybe this just quite special because of its simplicity.

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