Companion matrices, permutations and lattice ideals
Nsibiet E. Udo, Praise Adeyemo
TL;DR
This work uncovers a structural link between reductions of companion matrices for a symmetric family of square-free binomial ideals and permutation matrices. By analyzing a fixed Lex order on a zero-dimensional ideal $I_n \subset R_n$, it shows that the reduced companion matrices $P_i$ act as permutation matrices on a monomial basis excluding the constant term, with their actions forming a commuting family. The authors identify a finite abelian group $G_n \cong C_2^{n-2} \times C_{2n-4}$ generated by these permutations and interpret the $P_i$ as left-regular representations on the group algebra $F[G_n]$, yielding explicit cycle structures and orders. They then connect this group to a lattice ideal $I_{L_n}$, compute the Smith normal form of the relation matrix, and establish an isomorphism $R_n/I_{L_n} \cong F[G_n]$, clarifying how multiplication by $x_j$ corresponds to left multiplication by $g_j$ in the group algebra. This provides a concrete bridge between binomial ideals, permutation representations, and lattice-algebra structures with potential applications in combinatorial commutative algebra and Hilbert-scheme studies.
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.