Hecke and Artin monoids and their homomorphisms

TL;DR

This paper systematically studies homomorphisms of Hecke and Artin monoids, revealing new classes of homomorphisms that preserve parabolic elements and extend to covering monoids, enriching the algebraic theory.

Contribution

It introduces a comprehensive framework for homomorphisms of Hecke and Artin monoids, including many new types with specific properties.

Findings

Parabolic projections of Hecke monoids respect all parabolic elements.

Many homomorphisms of Hecke monoids lift to covering Artin monoids.

A large, mostly new, class of homomorphisms of Artin monoids is identified.

Abstract

The aim of the present work is to systematically study homomorphisms of Hecke and Artin monoids and thus to develop their comprehensive theory. Our original motivation was the striking observation that parabolic projections of Hecke monoids respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new.