# Hecke algebras of simply-laced type with independent parameters

## TL;DR

This paper investigates the structure and representations of Hecke algebras associated with simply-laced Coxeter systems with independent parameters, revealing their irreducible and projective modules, quivers, and conditions for finite representation type.

## Contribution

It constructs the irreducible and projective indecomposable representations of these Hecke algebras and analyzes their quivers and representation types, extending understanding of their module categories.

## Key findings

- Explicit construction of irreducible representations
- Determination of when the algebra has finite representation type
- Decomposition formulas for induced and restricted representations

## Abstract

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.