# Sage or Magma Implementation of Nilpotent Orbit Varieties

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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I added the tag software. Hoping it might attract a Magma or Sage expert. – Neil Hoffman May 16 '13 at 19:10
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. – Victor Protsak May 16 '13 at 21:10
@Victor Protsak: What do you mean by "Macaulay seems more likely to have something of this kind." Any references you could point me to? – Samuel Reid May 17 '13 at 19:10
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. – Victor Protsak May 23 '13 at 22:39

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.)