On the Algebra Generated by Three Commuting Matrices: Combinatorial Cases

Abstract

Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting $d \times d$ matrices over a field has dimension at most $d$. It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.