On combinatorial algebras generated by three commuting matrices
Ron Cherny, Tam An Le Quang, Matthew Satriano
TL;DR
The work tackles whether the dimension bound for the algebra generated by three pairwise commuting matrices holds by building a robust combinatorial framework of $S$-modules, skew shapes, and glueing data. It introduces floor plans and bottom-slice reductions to systematically reduce potential counterexamples to minimal configurations. A central result shows that right-free floor plans cannot realize counterexamples, supported by a two-dimensional resolution that enforces height-function constraints. Collectively, the authors prove a BIGMAIN-type statement for a broad class of combinatorial modules, indicating the dimension bound is preserved in these structured cases and laying a path toward a full resolution for $n=3$.
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.