On combinatorial algebras generated by three commuting matrices

2025/11/27 by Ron Cherny, Cherny, Ron, Tam An Le Quang +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2511.22092

Abstract

Motzkin and Taussky (and independently, Gerstenhaber) proved that the unital algebra generated by a pair of commuting d× d matrices over a field has dimension at most d. Since then, it has remained an open problem to determine whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for combinatorially-motivated classes of such triples.

Citations

• On the algebra generated by three commuting matrices: combinatorial cases
• Components and singularities of Quot schemes and varieties of commuting matrices
• New classes of examples satisfying the three matrix analog of\n Gerstenhaber's theorem
• Commuting matrices, and modules over Artinian local rings
• On varieties of commuting nilpotent matrices

Related

• On the algebra generated by three commuting matrices: combinatorial cases
• New classes of examples satisfying the three matrix analog of\n Gerstenhaber's theorem
• On the dimension of the algebra generated by two positive semi-commuting\n matrices
• A computing strategy and programs to resolve the Gerstenhaber Problem for commuting triples of matrices
• Commuting matrices, and modules over Artinian local rings