Jordan Type stratification of spaces of commuting nilpotent matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916587389321216 |
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| author | Boij, Mats Iarrobino, Anthony Khatami, Leila |
| author_facet | Boij, Mats Iarrobino, Anthony Khatami, Leila |
| contents | An $n\times n$ nilpotent matrix $B$ is determined up to conjugacy by a partition $P_B$ of $n$, its Jordan type given by the sizes of its Jordan blocks. The Jordan type $\mathfrak D(P)$ of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type $P$ is stable - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions $\mathfrak D^{-1}(Q)$ having a given stable partition $Q$ as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions $Q$ having $\ell$ parts: it was proven recently by J.~Irving, T. Košir and M. Mastnak.
Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in $\mathfrak D^{-1}(Q)$, when $Q$ is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jordan Type stratification of spaces of commuting nilpotent matrices Boij, Mats Iarrobino, Anthony Khatami, Leila Commutative Algebra Combinatorics Primary: 15A27, Secondary: 05A17, 13E10, 14A05, 15A20 An $n\times n$ nilpotent matrix $B$ is determined up to conjugacy by a partition $P_B$ of $n$, its Jordan type given by the sizes of its Jordan blocks. The Jordan type $\mathfrak D(P)$ of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type $P$ is stable - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions $\mathfrak D^{-1}(Q)$ having a given stable partition $Q$ as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions $Q$ having $\ell$ parts: it was proven recently by J.~Irving, T. Košir and M. Mastnak. Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in $\mathfrak D^{-1}(Q)$, when $Q$ is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable $Q$. |
| title | Jordan Type stratification of spaces of commuting nilpotent matrices |
| topic | Commutative Algebra Combinatorics Primary: 15A27, Secondary: 05A17, 13E10, 14A05, 15A20 |
| url | https://arxiv.org/abs/2409.13553 |