Some Combinatorial Cases of the Three Matrix Analog of Gerstenhaber’s Theorem (Research)

Let k be a field. By a theorem of Gerstenhaber from the early 1960s, the unital k-algebra generated by two pairwise commuting d × d matrices with entries in k is a finite dimensional k-vector space of dimension at most d. The analog of this theorem for four or more pairwise commuting matrices is false. The three matrix version remains open. In this paper, we use combinatorial and commutative-algebraic methods to prove that the three matrix analog of Gerstenhaber’s theorem holds for some infinite families of examples, each of which is combinatorial in nature.