Dual standard monomial theoretic and canonical bases for type \(A\)
type: Item
instance of: scholarly article
title: Dual standard monomial theoretic and canonical bases for type \(A\)
zbMATH Open document ID: 1039.17020
published in: Journal of Algebra
publication date: 2002-12-16T00:00:00Z
review text: Let \(U_q^-(\mathfrak g)\) be the negative part of the quantized universal enveloping algebra constructed from a Cartan matrix associated to a complex semisimple Lie algebra \(\mathfrak g\) of type \(A\). Let \(\lambda\) be a dominant integral weight and \(V(\lambda)\) the irreducible \(U_q^-(\mathfrak g)\)-module with highest weight \(\lambda\). In this paper, the authors show that the duals of the standard monomial theoretic bases for various \(V(\lambda)\) patch together to give a so-called dual standard monomial theoretic basis for \(U_q^-(\mathfrak g)\). This basis lives in the crystal lattice and the image modulo \(q\) of a basis element is the corresponding standard tableau thought of as a crystal. The main result is that the transformation matrix between this basis and the canonical basis is unipotent upper triangular with respect to a natural order on the set of standard tableaux. And all this holds over the integral forms. Finally, they compute explicitly the dual standard monomial theoretic basis for \(U_q^-(sl_3)\).
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1846015
zbMATH Keywords: canonical basis
zbMATH Keywords: quantized universal enveloping algebra
zbMATH Keywords: crystals
zbMATH Keywords: monomials
author: K. N. Raghavan
author: Parameswaran Sankaran
MaRDI profile type: Publication
full work available at URL: jabr.2001.9119
OpenAlex ID: W1965942567
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DOI: 10.1006/JABR.2001.9119
description: scientific article; zbMATH DE number 1846015