On combinatorial algebras generated by three commuting matrices

TL;DR

This paper investigates the dimension of algebras generated by three commuting matrices, providing new results for specific combinatorially-motivated classes, extending classical results known for pairs of matrices.

Contribution

It offers the first results confirming the dimension bounds for triples of commuting matrices in certain combinatorial cases, advancing understanding of algebraic structures generated by multiple matrices.

Findings

Dimension bounds established for specific classes of three commuting matrices

Extension of classical pairwise commuting matrix results to triples

New combinatorial methods for analyzing matrix-generated algebras

Abstract

Motzkin and Taussky (and independently, Gerstenhaber) proved that the unital algebra generated by a pair of commuting $d\times d$ 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.