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Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose Jordan form corresponds to the partition $\lambda$. Consider the variety $X_{A}$ of pairs of commuting nilpotent matrices which also commute with $A$:

$$X_{A}=\{(B,C)\in C_{\mathfrak{gl}_n}(A)^{\times2}\mid B^n=C^n=0,\;BC=CB\}$$

I'm interested in computing the dimension of $X_{A}$. My initial computations with Magma suggest the following conjecture:

If $s_i=|\{j|\lambda_j\ge i\}|$ then$$\dim X_{A}=n+\sum s_i^2-m-1$$

(The program I've written in Magma can't handle partitions of $\ge4$ parts, so although all computations thus far agree with the above formula, the evidence is far from conclusive)

How might I compute the dimension of $X_{A}$? Any general advice or techniques used in computing dimension?

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    $\begingroup$ Interesting question. I will try and think about this. In the meantime, here are some comments: You can decompose $k^n$ as $V_1\oplus\ldots\oplus V_m$, with $\dim V_i=s_i$. The map $A$ sends $V_i$ injectively to $V_{i+1}$. You may choose bases for the $V_i$ such that the matrix for $A$ as a map from $V_i$ to $V_{i+1}$ is an $s_i\times s_{i+1}$ matrix $\begin{pmatrix}I|0\end{pmatrix}$. I think given this, you can say a lot about the structure of $B$ and $C$. I would then hope to describe the variety that you're interested in. $\endgroup$ Mar 13, 2014 at 17:14
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    $\begingroup$ I'd say that this is a no so trivial question. Namely, if you omit the "commute with $A$" condition, you get the variety of commuting nilpotent matrices. Its dimension is $n^2-1$, which is not obvious (see "The Variety of Pairs of Commuting Nilpotent Matrices is Irreducible" by Volodia Baranovsky). $\endgroup$ Mar 15, 2014 at 16:28
  • $\begingroup$ You can refer a paper of Ngo and Sivic for partial solution arxiv.org/abs/1308.4438 $\endgroup$
    – NN guest
    Apr 19, 2014 at 12:53
  • $\begingroup$ In general, it is still open problem. $\endgroup$
    – NN guest
    Apr 19, 2014 at 12:54

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