Dual braid monoids, Mikado braids and positivity in Hecke algebras

TL;DR

This paper explores the geometric and algebraic properties of rational permutation braids in Artin-Tits groups, showing their positivity in Hecke algebra expansions and proposing conjectures for broader types.

Contribution

It provides a geometric characterization of rational permutation braids in types A and B, and demonstrates their positivity properties in Hecke algebra representations, with conjectures extending to type D.

Findings

Rational permutation braids are characterized geometrically in types A and B.

In spherical types other than D, simple elements of the dual braid monoid are rational permutation braids.

These properties imply positivity in Hecke algebra and Temperley-Lieb algebra expansions.

Abstract

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type $A_n$, it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.