Hecke monoids, their homomorphisms and parabolicity

TL;DR

This paper investigates homomorphisms of Hecke monoids, focusing on parabolic and injective types, providing classifications and insights into their structure and interrelations.

Contribution

It classifies locally injective homomorphisms between classical Hecke monoids and explores the structure of parabolic homomorphisms, revealing new understanding of their properties.

Findings

Classified all locally injective connected homomorphisms between classical Hecke monoids.

Identified a wide range of parabolic homomorphisms within Hecke monoids.

Described, to some extent, all homomorphisms between Hecke monoids.

Abstract

We study homomorphisms of Hecke monoids, notably parabolic homomorphisms, which map parabolic elements to parabolic elements, and injective ones. The importance of the first class stems from the fact that parabolic elements form a rather mysterious submonoid of the Hecke monoid, and we found a plethora of parabolic homomorphisms.Concerning injective ones, as a first step towards their classification, we classified all locally injective connected homomorphisms between Hecke monoids of classical types and expect all of them to be injective. As a surprising byproduct of our study of parabolic and injective homomorphisms we described, to some extent, all homomorphisms between Hecke monoids.