Monomial irreducible \(\mathfrak{sl}_n(\mathbb C)\)-modules
type: Item
instance of: scholarly article
title: Monomial irreducible \(\mathfrak{sl}_n(\mathbb C)\)-modules
zbMATH Open document ID: 1273.17014
published in: Bulletin of the Iranian Mathematical Society
publication date: 2012-11-16T00:00:00Z
full work available at URL: 87820110313.pdf
review text: Let \(L\) be a finite dimensional complex simple Lie algebra and \(H\) a Cartan subalgebra of \(L\). Given integral dominant weight \(\lambda\), let \(L(\lambda)\) denote the finite dimensional irreducible representation of \(L\) with highest weight \(\lambda\). The aim of the article under review is to find an ordered basis for \(L(\lambda)\) with respect to which the action of \(L\) on \(L(\lambda)\) can be represented by matrices. To the aim the author gives a suitable basis for \(L(\lambda)\) for \(L = \mathfrak{sl}_n(\mathbb{C})\) and for \(\lambda\) monomial weights. A simple formula for the action of Chevalley generators for \(\mathfrak{sl}_n(\mathbb{C})\) on the basis elements is also given. To obtain such a basis the fact that \(L(\lambda)\) is isomorphic to a symmetry class of tensors (see, \textit{A. Madadi} and \textit{M. Shahryari} [Linear Multilinear Algebra, 56, 517--541 (2008; Zbl 1147.15024)]) plays a central role.
reviewed by: Toshihisa Kubo
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 6108179
zbMATH Keywords: symmetric group
zbMATH Keywords: character theory
zbMATH Keywords: representations of Lie algebras
zbMATH Keywords: symmetry classes of tensors
author: Mohammad Shahryari
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6108179