On the combinatorics of Kostant's partition function
type: Item
instance of: scholarly article
title: On the combinatorics of Kostant's partition function
zbMATH Open document ID: 0603.17005
DOI: 10.1016/0021-8693(85)90034-1
author: Robert W. Deckhart
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The author proves the following technical theorem. Let R be a reduced root system with base \(B=\{\alpha_ 1,...,\alpha_ n\}\) and let \(- C_{ji}=<\alpha_ j,\alpha_ i>\), the Cartan integer. Theorem. Let \(t_ i\) (1\(\leq i\leq n)\) be nonnegative integers. Then \(\sum t_ i \alpha_ i\) can be written as a sum of positive nonsimple roots with nonnegative coefficients iff \(t_ i\leq \sum_{j\neq i}C_{ji} t_ j\) for every \(i=1,...,n.\) One direction in the proof is quite delicate using a construction which handles the type \(G_ 2\) case separately from the other types. As an application of this theorem, there is a result in representation theory. Let U be the universal enveloping algebra of a complex semisimple Lie group and let V(\(\Lambda)\) be an irreducible U-module of highest weigth \(\Lambda\). The result is roughly that the multiplicity of any weight space in V(\(\Lambda)\) is determined by the multiplicities of the allbracket ones.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 3973106
zbMATH Keywords: Kostant's partition function
zbMATH Keywords: sum of simple roots
zbMATH Keywords: sum of positive nonsimple roots
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: reduced root system
zbMATH Keywords: representation
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: complex semisimple Lie group
zbMATH Keywords: highest weigth
zbMATH Keywords: multiplicity
zbMATH Keywords: weight space
reviewed by: Thomas A. Farmer
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90034-1
OpenAlex ID: W2088230859
cites work: Q5580319
cites work: Modular maximal vectors and the Kostant cone
cites work: Introduction to Lie Algebras and Representation Theory
description: scientific article; zbMATH DE number 3973106