Schur functions and affine Lie algebras
type: Item
instance of: scholarly article
title: Schur functions and affine Lie algebras
zbMATH Open document ID: 0989.17018
author: Bernard Leclerc
author: Séverine Leidwanger
published in: Journal of Algebra
publication date: 1999-04-26T00:00:00Z
review text: It is known that irreducible highest weight representations of \(a_{\infty}\), an infinite rank affine Lie algebra, can be realized on the space \(Sym\) of symmetric functions in countably many variables. A canonical basis of weight vectors of \(Sym\) is given by Schur's S-functions. Analogously, the infinite rank affine Lie algebra \(b_{\infty}\), an infinite-dimensional analogue of the Lie algebra \(so_{2n+1}\), has representations contained again in \(Sym\) and a canonical basis of weight vectors given by Schur's P-functions. The authors obtain an explicit intertwining operator which allows them to compute relations between S-functions and P-functions. They also show how to realize the representations of the affine Lie algebra \(A_1^{(1)}\) both in the homogeneous and in the principal picture on \(Sym\) and how to use this situation to get new and different relations between S-functions and P-functions.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1247570
zbMATH Keywords: Schur functions
zbMATH Keywords: affine Lie algebras
zbMATH Keywords: intertwining operator
describes a project that uses: Maple
describes a project that uses: SCHUR
describes a project that uses: ACE
MaRDI profile type: Publication
full work available at URL: jabr.1998.7453
OpenAlex ID: W2055735370
Wikidata QID: Q115395885
cites work: Weight vectors of the basic A1(1)-module and the Littlewood-Richardson rule
cites work: Splitting the square of a Schur function into its symmetric and antisymmetric parts
cites work: Transformation groups of soliton equations. IV: A new hierarchy of soliton equations of KP-type
cites work: The maximal subgroups of the classical groups
cites work: Basic representations of affine Lie algebras and dual resonance models
cites work: Q3952291
cites work: Combinatorial description of the Fock representation of the affine Lie algebra \(\text{go}(\infty)\)
cites work: Solitons and infinite dimensional Lie algebras
cites work: Vertex operators, symmetric functions, and the spin group \(\Gamma_ n\)
cites work: SchurQ-functions and cohomology of isotropic Grassmannians
cites work: A Determinantal Formula for Skew Q -Functions
cites work: Q3693664
cites work: Q5286566
cites work: Hecke algebras at roots of unity and crystal bases of quantum affine algebras
cites work: Fonctions P de Schur et représentations d'algèbres de Lie affines
cites work: Q4887220
cites work: q-Deformed Fock spaces and modular representations of spin symmetric groups
cites work: Construction of the affine Lie algebra \(A^{(1)}_1\)
cites work: Modular representations of symmetric groups
cites work: Q3947818
cites work: The Spin Representation of the Symmetric Group
cites work: Some combinatorial results involving shifted Young diagrams
cites work: Basic representations of A2l(2)and Dl+1(2)and the polynomial solutions to the reduced BKP hierarchies
cites work: Frobenius symbols for partitions and degrees of spin characters.
cites work: Q4299617
cites work: Enumerative geometry of degeneracy loci
cites work: A generalization of the Macdonald-You formula
cites work: THE TENSOR ALGEBRA OF THE IDENTITY REPRESENTATION AS A MODULE OVER THE LIE SUPERALGEBRAS $ \mathfrak{Gl}(n,\,m)$ AND $ Q(n)$
cites work: On Schur's \(Q\)-functions and the primitive idempotents of a commutative Hecke algebra
cites work: Q5001957
cites work: Q4011449
cites work: On some identities of Schur \(Q\)-functions
DOI: 10.1006/JABR.1998.7453
description: scientific article; zbMATH DE number 1247570