# Lattice extensions of Hecke algebras

## TL;DR

This paper explores lattice extensions of Hecke algebras associated with finite reflection groups, generalizing known results and establishing their structure and symmetry properties, especially for Coxeter groups.

## Contribution

It introduces a general framework for lattice extensions of Hecke algebras and proves their structure and symmetry properties, extending previous results from symmetric groups to all finite reflection groups.

## Key findings

- Algebras are symmetric for Coxeter groups.
- Generalization of the structure theorem from symmetric groups to all reflection groups.
- Connection to diagram algebras of braids and ties.

## Abstract

We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when $W$ is a Coxeter group, and in general under the trace conjecture of Brou\'e, Malle and Michel.