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Mathematics > Combinatorics
Title: A Proof of the Box Conjecture for Commuting Pairs of Matrices
(Submitted on 27 Mar 2024 (v1), last revised 2 Apr 2024 (this version, v2))
Abstract: 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.
Submission history
From: Mitja Mastnak [view email][v1] Wed, 27 Mar 2024 13:55:04 GMT (32kb,D)
[v2] Tue, 2 Apr 2024 21:30:09 GMT (33kb)
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