A divisibility result on combinatorics of generalized braids

TL;DR

This paper investigates the algebraic structure of positive braids in Coxeter groups, proving divisibility properties of adjacency matrices for types B and others, using Hopf algebras, and highlighting differences among types.

Contribution

It establishes a divisibility result for the characteristic polynomials of adjacency matrices in Coxeter braid groups, extending known results and employing Hopf algebra techniques.

Findings

Divisibility of characteristic polynomials for type B adjacency matrices

Use of Hopf algebra based on signed permutations

Differences observed in type D and other Coxeter types

Abstract

For every finite Coxeter group $\Gamma$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $\Gamma$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(\Gamma)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. The other Coxeter types ($I$, $E$, $F$ and $H$) are also studied.