Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras
Year: 2026
Type: preprint
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 $a,c\ge2$ 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 $14\le n\le52$ and $93\le n\le114$, 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.
Source: arXiv (Cornell University)
Author: Nowak-Kępczyk, Małgorzata
Institution: John Paul II Catholic University of Lublin
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160278230