For a given partition $[n_{1},...,n_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O_{[n_{1},...,n_{k}]}$ in $\mathfrak{gl}(N)$ which can be represented by a set of polynomial equations relating the conditions on matrices in $\mathfrak{gl}(N)^{\text{nilp}}$. I was wondering if anyone has implemented the computation of nilpotent orbit varieties in Sage or Magma, because otherwise I am going to make my own code for doing so. If you happen to know of any references, that would help!
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$\begingroup$ I added the tag software. Hoping it might attract a Magma or Sage expert. $\endgroup$– Neil HoffmanCommented May 16, 2013 at 19:10
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$\begingroup$ Can you, please, clarify what kind of operations or properties you are interested in? Also, I am not an expert, but Macaulay seems more likely to have something of this kind. $\endgroup$– Victor ProtsakCommented May 16, 2013 at 21:10
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$\begingroup$ @Victor Protsak: What do you mean by "Macaulay seems more likely to have something of this kind." Any references you could point me to? $\endgroup$– Samuel ReidCommented May 17, 2013 at 19:10
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$\begingroup$ I still don't understand what you are trying to accomplish, but Macaulay was written by commutative algebra/algebraic geometry researchers and implements a lot more stuff relevant to these subjects than SAGE. I've added a theoretical answer below. $\endgroup$– Victor ProtsakCommented May 23, 2013 at 22:39
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I don't know how this would help "constructing local weak Neron models for the varieties given as output of the program", but the defining equations of the nilpotent varieties are known. They were conjectured by Tanisaki and proved by Weyman in
The equations of conjugacy classes of nilpotent matrices, Invent. Math. 98:2 (1989).
Weyman's later book, Cohomology of vector bundles and syzygies, contains a complete exposition.
The easiest way to describe the equations on a matrix $A$ in the nilpotent variety is by saying that the characteristic polynomial of $A$ is equal to $\lambda^N$ and for $1\leq k\leq N-1$, the order $k$ minors of $\lambda I_N-A$ are divisible by $\lambda^{d_k}$, where $d_k$ is an integer given by a simple formula in terms of the parts of the corresponding partition.
(These generators for the defining ideal are $Ad$-invariant but not minimal, and there is an alternative system, also non-minimal, whose definition involves the inverse matrix $(\lambda I_N -A)^{-1}$, that may be better computationally.)
answered May 23, 2013 at 22:36