Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras

Abstract

We study equality cases in Laffey's lower bound for local maximal commutative subalgebras of matrix algebras with cube-zero radical. We introduce a stacking operation that combines algebras with common outer Loewy layers while preserving maximal commutativity. This yields standard equality bricks $B_{a,c}=(a,ac+1,c;a+c)$ for all $a,c\ge 2$ and reduces further constructions to smaller building blocks. We establish the existence of all minimal equality bricks on the balanced branch and obtain sharp realizations throughout the $(4,5)$-sector. We also construct the first nontrivial nearly balanced root over arbitrary fields. We determine the balanced--nearly-balanced envelope of the numerical Laffey profile and give an arithmetic criterion for preservation of sharpness under stacking. As a consequence, the profile is attained in every matrix dimension $14\le n\le 52$ and $93\le n\le 114$, as well as along five infinite families over arbitrary fields. The main existence and interval results are proved in characteristic zero; the field-independent constructions are stated explicitly.

arXiv commentSubstantially revised and reorganized version, with strengthened proofs and additional results. 32 pages, 1 figure, 1 table