Nonexistence of twisted Hecke algebras
type: Item
instance of: scholarly article
title: Nonexistence of twisted Hecke algebras
zbMATH Open document ID: 0862.20008
author: Akihiko Gyoja
published in: Journal of Mathematics of Kyoto University
publication date: 1997-04-07T00:00:00Z
review text: Let \((W,S)\) be a Coxeter system with a finite set \(S\neq\emptyset\). The group ring \(\mathbb{C}[W]\) can be deformed into the twisted group ring \(\mathbb{C}[W]'\) by a cocycle or to the \(q\)-deformation of the group ring \(H[q,w]\) [the Iwahori Hecke algebra] where \(q=\{q_w\}_{w\in W}\) is a family of non-zero complex numbers such that \(q_xq_y=q_{xy}\) if \(\ell(xy)=\ell(x)+\ell(y)\) where \(\ell\) is the length function on \(W\). The algebra \(H[q,w]\) has \(\mathbb{C}\)-basis \(\{T_w\}_{w\in W}\) and multiplication given by \[ T_sT_w=\begin{cases} T_{sw} &\text{if \(sw>w\)}\\ q_sT_{sw}+(q_{s^{-1}})T_w &\text{if \(sw<w\)}\end{cases} \] where \(\leq\) is the Bruhat order. The author proves that, in a sense, the \(q\)-deformation of the twisted group ring does not exist.
reviewed by: Ahmed A. Khammash
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 950579
zbMATH Keywords: Coxeter systems
zbMATH Keywords: twisted group rings
zbMATH Keywords: \(q\)-deformations
zbMATH Keywords: Iwahori Hecke algebras
zbMATH Keywords: Bruhat order
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518604
OpenAlex ID: W1584990511
DOI: 10.1215/KJM/1250518604
description: scientific article; zbMATH DE number 950579