On complexes relating the Jacobi-Trudi identity with the Bernstein- Gelfand-Gelfand resolution. II
type: Item
instance of: scholarly article
title: On complexes relating the Jacobi-Trudi identity with the Bernstein- Gelfand-Gelfand resolution. II
zbMATH Open document ID: 0794.20052
DOI: 10.1016/0021-8693(92)90039-O
author: Kaan Akin
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: In two earlier papers [Part I, J. Algebra 117, No. 2, 494-503 (1988; Zbl 0668.20033); Contemp. Math. 88, 209-217 (1989; Zbl 0721.20023)] the author constructed complexes \(Y(\lambda)\), \(Y'(\lambda)\) of \(GL_ n(K)\)- modules, where \(K\) is a field of characteristic zero and \(\lambda= (\lambda_ 1, \lambda_ 2,\dots, \lambda_ m)\) is a sequence of non- negative integers. The complex \(Y(\lambda)\) gives a realization of the Jacobi-Trudi identity. In this paper he describes the boundary maps of \(Y(\lambda)\) and \(Y'(\lambda)\) in terms of certain Shapovalov elements in the enveloping algebra \(U({\mathfrak N}^ -)\), where \({\mathfrak N}^ -\) is the Lie algebra of strictly lower triangular \(m\times m\) matrices. He also describes certain subcomplexes of \(Y(\lambda)\) and \(Y'(\lambda)\) that can be obtained recursively from complexes associated with sequences of positive integers whose sum equals \(m\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 130489
zbMATH Keywords: complexes of \(GL_ n(K)\)-modules
zbMATH Keywords: Jacobi-Trudi identity
zbMATH Keywords: boundary maps
zbMATH Keywords: Shapovalov elements
zbMATH Keywords: subcomplexes
reviewed by: Bhama Srinivasan
MaRDI profile type: Publication
cites work: On complexes relating the Jacobi-Trudy identity with the Bernstein- Gelfand-Gelfand resolution
cites work: Q5752745
cites work: Characteristic-free representation theory of the general linear group. II: Homological considerations
cites work: Q4107261
cites work: On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra
cites work: Resolutions, dual pairs, and character formulas
description: scientific article; zbMATH DE number 130489