A Proof of the Box Conjecture for Commuting Pairs of Matrices

2024/03/27 by J. Irving, Irving, J., Tomaž Košir +3
Computer Science · Engineering · Physics and Astronomy · #05A17 #13E10 #14A25 #15A21 #15A27 #Advanced Mathematical Theories and Applications #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2403.18574

openalex publication_date 2024/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Jordan Type stratification of spaces of commuting nilpotent matrices
• On pairs of commuting nilpotent matrices
• The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix
• Pairs of commuting nilpotent matrices, and Hilbert function
• Jordan structures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size