On the Coefficients of the Permanent and the Determinant of a Circulant Matrix. Applications

TL;DR

This paper investigates the relationship between the number of summands in the permanent and determinant of circulant matrices, proving they are equal if and only if the size is a prime power, with applications to algebraic properties.

Contribution

It establishes a precise condition linking the coefficients of the permanent and determinant of circulant matrices, and applies this to algebraic geometry.

Findings

d(N) equals p(N) if and only if N is a prime power

Provides a criterion for when the permanent and determinant coefficients coincide

Applies the result to monomial ideals and the Weak Lefschetz property

Abstract

Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.