Sur une conjecture de Dehornoy

TL;DR

This paper proves a conjecture by Dehornoy that the characteristic polynomial of a specific permutation matrix divides the polynomial of the next size, revealing a divisibility property in permutation matrices.

Contribution

It establishes a divisibility relation between characteristic polynomials of matrices defined by descent properties of permutations, confirming Dehornoy's conjecture.

Findings

Proves the divisibility of characteristic polynomials P_n(x) dividing P_{n+1}(x)

Confirms a conjecture by P. Dehornoy

Provides insight into the structure of matrices related to permutation descents

Abstract

Let M_n be the n! * n! matrix indexed by permutations of S_n, defined by M_n(sigma,tau)=1 if every descent of tau^{-1} is also a descent of sigma, and M_n(sigma,tau)=0 otherwise. We prove the following result, conjectured by P. Dehornoy: the characteristic polynomial P_n(x)=|xI-M_n| of M_n divides P_{n+1}(x) in Z[x].