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arXiv · 2403.18574

A Proof of the Box Conjecture for Commuting Pairs of Matrices

J. Irving·T. Košir·M. Mastnak
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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.

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J. Irving, T. Košir, M. Mastnak. 2024-03-27. A Proof of the Box Conjecture for Commuting Pairs of Matrices. https://arxiv.org/abs/2403.18574

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