Companion matrices, permutations and lattice ideals

TL;DR

This paper uncovers a new link between companion matrix reductions, permutation matrices, and lattice ideals, revealing group-theoretic structures in algebraic geometry.

Contribution

It introduces a novel connection between companion matrix reductions and permutation matrices within the context of lattice ideals, expanding understanding of their algebraic properties.

Findings

Reduced companion matrices produce permutation matrices satisfying group relations

Connections established between matrix reductions and lattice ideals

Characterization of groups generated by these transformations

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.