Good bases for G-modules
type: Item
instance of: scholarly article
title: Good bases for G-modules
zbMATH Open document ID: 0704.20035
DOI: 10.1007/BF00181464
author: Olivier Mathieu
published in: Geometriae Dedicata
publication date: 1990-01-01T00:00:00Z
review text: Let G be a semi-simple algebraic group over \({\mathbb{C}}\). Let T be a maximal torus in G and B a Borel subgroup, \(B\supset T\). Let R be the root system of G relative to T and S be the system of simple roots of R relative to B. For \(\beta\in R\), let \(X_{\beta}\) denote the element of the Chevalley basis of Lie(G) corresponding to \(\beta\). Let V be a G-module and let \(\lambda\) be any dominant weight, say \(\lambda =\sum^{n}_{i=1}a_ i\omega_ i\), where \(n=rank(G)\) and \(\{\omega_ i\), \(1\leq i\leq n\}\) is the set of fundamental weights, and \(a_ i\in {\mathbb{Z}}^+\). Let \(V(\lambda)=\{v\in V|\) \(X_{\alpha_ i}^{(a_ i+1)}v=0\), \(1\leq i\leq n\}\). Let \({\mathcal B}\) be a basis of V. For a dominant weight \(\lambda\), let \({\mathcal B}_{\lambda}={\mathcal B}\cap V(\lambda)\). The basis \({\mathcal B}\) is said to be good if \({\mathcal B}_{\lambda}\) is a basis of V(\(\lambda\)), for any dominant weight \(\lambda\). It was conjectured independently by Baclawski and Gelfand- Zelevinsky that every G-module V has a good basis. The author proves the above conjecture. One of the motivations for considering good bases is to describe the Clebsch-Gordan coefficients (in the tensor product decomposition); for example, if \({\mathcal B}=\{b_ i\}\) and \({\mathcal D}=\{d_ j\}\) are good bases for two G-modules V, W respectively, then \(\{b_ i\otimes d_ j\}\) is a good basis for \(V\otimes W\). This paper makes an important contribution to the Representation Theory of semi- simple algebraic groups.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4154729
zbMATH Keywords: semi-simple algebraic group
zbMATH Keywords: maximal torus
zbMATH Keywords: Borel subgroup
zbMATH Keywords: system of simple roots
zbMATH Keywords: dominant weight
zbMATH Keywords: fundamental weights
zbMATH Keywords: good bases
zbMATH Keywords: Clebsch- Gordan coefficients
zbMATH Keywords: tensor product decomposition
zbMATH Keywords: G-modules
reviewed by: V. Lakshmibai
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181464
OpenAlex ID: W2010855929
Recommended article: Q3483464
Recommended article: 𝐺-modules with a convex base
Recommended article: Based modules and good filtrations in algebraic groups
Recommended article: Some studies of Gröbner bases for modules and applications
Recommended article: Q4472961
Recommended article: Gröbner bases of modules over reduction rings
Recommended article: Bitableaux bases for some Garsia-Haiman modules and other related modules
Recommended article: Q3835556
Recommended article: Gröbner bases and primary decomposition of modules
Recommended article: Gröbner bases for some toric rings
description: scientific article; zbMATH DE number 4154729