Generalization of Jordan Decomposition for Several Commuting Operators

Question

Recently I became curious about the following question:

Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: describe the structure of $V$ as a module over $k[x_1, \cdots, x_n]$ where $x_i$ acts by $A_i$.

Since $V$ has a finite length as $k[x_1, \cdots, x_n]$-module, after we quotient by annihilator of $V$, we have a module over artinian ring (geometrically speaking, support is discrete set), so $V$ is isomorphic to direct sum of it's localizations at prime ideals (all primes are maximal in this case). Is there a description of each of the local components? We can assume that $k$ is algebraically closed, so that Nullstellensatz might help.

Thanks for your replies.

Answer 1

It is well known that representation theory of a (even commutative) Artinian $k$-algebra $R$ can be wild (meaning that one can embed $\mod(A)$ into $\mod(R)$ for any finite-dimensional, not necessarily commutative, $k$-algebra $A$). The easiest example is $k[x,y]/(x^2,xy^2,y^3)$, see Example 4 in this paper and the references there.

Answer 2

Let me elaborate on Johannes Ebert's comment.

Since you are dealing with commuting operators, you can find a basis for $V$ so that the matrices for $A_1,\ldots,A_n$ are in Jordan normal form. So, $V$ decomposes as $V=V_1\oplus\cdots\oplus V_r$, where for each $i=1,\ldots,r$, there exists scalars $a_{ij}$, $j=1,\ldots,n$, such that $(x_j-a_{ij})^Nv=0$ for all $v\in V_i$ and $N\gg0$. Moreover, we may assume that each $V_i$ is indecomposable.

Now, we may assume $r=1$, and let $a_j$ be the generalized eigenvalues for the action of $x_j$ on $V$. By replacing $x_j$ by $x_j-a_j$, we may assume all $a_j$ are 0, and the $x_j$ are nilpotent.

Now, the problem is to identify commuting families of $n$ nilpotent operators acting on a finite dimensional vector space, say of dimension $N$. To do this, for each subset $S\subset\{ 1,\ldots,N-1\}$, consider the matrix $e_S=\sum_{i\in S}e_{i,i+1}$. Then, we are looking for a collection of subsets $S_1,\ldots,S_n$ such that $e_{S_i}e_{S_j}=e_{S_j}e_{S_i}$. If we consider the special case where $S_i$ and $S_j$ are intervals, we see that we must have either $S_i=S_j$ or $S_i\cup S_j$ consists of two disjoint intervals. To generalize, each $S_i=\bigcup S_{ik}$, where each $S_{ik}$ is an interval, but no $S_{ik}\cup S_{il}$ is an interval (or $S_i=\emptyset$).

Answer 3

I haven't read this paper, but it looks relevant: Principal nilpotent pairs in a semisimple Lie algebra I. To quote from the MathSciNet review:

The author notes that the general problem of classifying $G$-orbits through arbitrary nilpotent pairs is wild; even the set of orbits of maximal dimension has no structure of algebraic variety.

Answer 4

I thought the result that this is a wild problem was due to Ringel. However I have not been able to find the reference. Here is a reference for pairs of nilpotent elements (without the assumption that they commute).

MR2376281 (2009d:16016) Ringel, Claus Michael ; Schmidmeier, Markus . Invariant subspaces of nilpotent linear operators. I. J. Reine Angew. Math. 614 (2008), 1--52.