On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\)
type: Item
instance of: scholarly article
title: On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\)
zbMATH Open document ID: 0888.20011
author: Susumu Ariki
published in: Journal of Mathematics of Kyoto University
publication date: 1998-06-07T00:00:00Z
review text: Let \(H_A\) be the Hecke algebra over the polynomial ring \(A=\mathbb Z[v_1,\dots,v_m,q,q^{-1}]\) defined by generators \(a_1,\dots,a_n\) and relations \((a_1-v_1)\cdots(a_1-v_m)=0\), \((a_i-q)(a_i+q^{-1})=0\) (\(2\leq i\leq n\)), \(a_1a_2a_1a_2=a_2a_1a_2a_1\), \(a_ia_j=a_ja_i\) \(\forall j\geq i+2\), \(a_ia_{i+1}a_i=a_{i+1}a_ia_{i+1}\) (\(2\leq i\leq n-1\)). This algebra is \(A\)-free and when specializing \(v_i\) \& \(q\) with \(v_i\in\mathbb C\) \& \(q\in\mathbb C^\times\) one gets \(H_\mathbb C\). Let \(u_n\) be the Grothendieck group of the category of \(H_\mathbb C\)-modules and let \(u=\bigoplus u_n\). The main aim of this paper is to show that the graded dual of \(u\) is a highest weight module of \(g(A_\infty)\) (resp. \(g(A^{(1)}_{r-1})\)) if \(q^2\) is not a root of unity (resp. a primitive \(r\)-th root of unity), and the dual basis of irreducible modules coincides with the canonical basis. The proof depends on Lusztig's theory of affine Hecke algebras and quantum groups, and Ginzburg's theory of affine Hecke algebras.
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1062466
zbMATH Keywords: Hecke algebras
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: highest weight modules
zbMATH Keywords: irreducible modules
zbMATH Keywords: affine Hecke algebras
zbMATH Keywords: quantum groups
reviewed by: Ahmed A. Khammash
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518452
OpenAlex ID: W1526024217
DOI: 10.1215/KJM/1250518452
description: scientific article; zbMATH DE number 1062466