Some canonical basis vectors in the basic \(U_q(\widehat{{\mathfrak sl}_n})\)-module
type: Item
instance of: scholarly article
title: Some canonical basis vectors in the basic \(U_q(\widehat{{\mathfrak sl}_n})\)-module
zbMATH Open document ID: 1021.17012
author: Joseph Chuang
author: Kai Meng Tan
published in: Journal of Algebra
publication date: 2003-10-08T00:00:00Z
review text: Let \(U_q(\widehat{sl_n})\) denote the quantized enveloping algebra associated to the affine Kac-Moody Lie algebra \(\widehat{sl_n}\). The basic \(U_q(\widehat{sl_n})\)-module has a canonical basis with favourable properties. \textit{A. Lascoux, B. Leclerc} and \textit{J.-Y. Thibon} [Commun. Math. Phys. 181, 205-263 (1996; Zbl 0874.17009)] gave a fast algorithm to compute this basis. In this paper, the authors give an explicit formula for a certain class of canonical basis vectors. This includes the entire canonical basis in a family of weight spaces of arbitrarily large dimension. This formula is computed via the wreath products of symmetric groups which arise in the proof of Broue's conjecture for certain blocks of the symmetric groups, see \textit{J. Chuang} and \textit{R. Kessar} [Bull. Lond. Math. Soc. 34, 174-184 (2002; Zbl 1033.20009)]. See also \textit{B. Leclerc} and \textit{H. Miyachi} [Represent. Theory 6, 290-312 (2002; Zbl 1030.17014)] for another formula for canonical basis elements in the basic \(U_q(\widehat{sl_n})\)-module.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1749651
zbMATH Keywords: canonical basis
zbMATH Keywords: Fock space
zbMATH Keywords: basic module
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: block theory
zbMATH Keywords: symmetric group
zbMATH Keywords: Broue's conjecture
zbMATH Keywords: wreath products
zbMATH Keywords: Specht module
zbMATH Keywords: partition
zbMATH Keywords: abacus
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
full work available at URL: jabr.2001.9030
OpenAlex ID: W1969618774
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DOI: 10.1006/JABR.2001.9030
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description: scientific article; zbMATH DE number 1749651