A Proof of the Box Conjecture for Commuting Pairs of Matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929301389049856 |
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| author | Irving, J. Košir, T. Mastnak, M. |
| author_facet | Irving, J. Košir, T. Mastnak, M. |
| contents | We prove the Box Conjecture for pairs of commuting nilpotent matrices, as formulated by Iarrobino et al [28]. This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool is the Burge correspondence between the set of all partitions and a set of binary words [15, 16]. For connection with the algebraic and geometric setup of matrices and orbits we employ some of Shayman's results on invariant subspaces of a nilpotent matrix [45, 46]. Our proof is valid over an arbitrary field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Proof of the Box Conjecture for Commuting Pairs of Matrices Irving, J. Košir, T. Mastnak, M. Combinatorics Commutative Algebra Algebraic Geometry 15A27, 05A17, 15A21, 13E10, 14A25 We prove the Box Conjecture for pairs of commuting nilpotent matrices, as formulated by Iarrobino et al [28]. This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool is the Burge correspondence between the set of all partitions and a set of binary words [15, 16]. For connection with the algebraic and geometric setup of matrices and orbits we employ some of Shayman's results on invariant subspaces of a nilpotent matrix [45, 46]. Our proof is valid over an arbitrary field. |
| title | A Proof of the Box Conjecture for Commuting Pairs of Matrices |
| topic | Combinatorics Commutative Algebra Algebraic Geometry 15A27, 05A17, 15A21, 13E10, 14A25 |
| url | https://arxiv.org/abs/2403.18574 |