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 Ba,c=(a,ac+1,c;a+c) for all a,c≥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≤ n≤52 and 93≤ n≤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.
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BibTeX
@misc{nowakkepczyk2026stack,
title = {{Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras}},
author = {Małgorzata Nowak-Kępczyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.01387},
eprint = {2605.01387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.01387},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras
AU - Małgorzata Nowak-Kępczyk
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2605.01387
UR - https://arxiv.org/abs/2605.01387
AB - 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≥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≤ n≤52 and 93≤ n≤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.
ER -