Companion matrices, permutations and lattice ideals

2025/11/09 by Udo, Nsibiet E., Adeyemo, Praise
Computer Science · Mathematics · #06B10 (Primary) 20K01 #13A15 #20D45 #20F28 (Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Group Theory (math.GR) #Polynomial and algebraic computation #Representation Theory (math.RT) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.2511.06555

openalex publication_date 2025/11/09 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

This paper investigates a novel connection between reductions of companion matrices associated with a symmetric family of certain binomial ideals in the coordinate ring of affine n-space and permutation matrices. Specifically, for fixed monomial orders, we observe that the reduced companion matrices yield permutation matrices satisfying group-theoretic relations. We explore the implications of these reductions, their connections to lattice ideals, and characterize the groups generated by these transformations.

Citations

• Ideals, representations and a symmetrised Bernoulli triangle
• Smith Normal Form in Combinatorics
• Automorphisms of finite Abelian groups

Related

• Smith forms of matrices in Companion Rings, with group theoretic and topological applications
• Double centralisers and annihilator ideals of Young permutation modules
• A Note on the Classification of Permutation Matrix
• Subalgebras of Matrix Algebras Generated by Companion Matrices
• Congruence lattices of ideals in categories and (partial) semigroups