Characteristic-free representation theory of the general linear group. II: Homological considerations
type: Item
instance of: scholarly article
title: Characteristic-free representation theory of the general linear group. II: Homological considerations
zbMATH Open document ID: 0681.20028
DOI: 10.1016/0001-8708(88)90027-8
author: Kaan Akin
author: David A. Buchsbaum
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This paper is a sequel to Part I [ibid. 58, 149-200 (1985; Zbl 0607.20021)], and gives proofs of some results announced there. Let F be a free module of rank n over a commutative ring R. The symmetric group \(\Gamma =\Gamma (r)\) on the set \(\{\) 1,2,...,r\(\}\) acts naturally on \(F^{\otimes r}\) and the algebra of \(\Gamma\)-endomorphisms of this \(\Gamma\)-module is the Schur algebra \(A=A_ R(n,r)\). Then A can be identified with the r-fold divided power algebra \(D_ r(E)\) where \(E=End_ R(F)\). The authors consider the family of ``shapes'' \(J=\cup_{n\geq 0}J_ n\), where \(J_ n\) is the set of all relative sequences \(\gamma =(\lambda_ 1,...,\lambda_{n+1})/(\mu_ 1,...,\mu_{n+1})\) where \((\lambda_ 1,...,\lambda_{n+1})\) and \((\mu_ 1,...,\mu_{n+1})\) are partitions and there exists a non- negative integer \(i<n+1\) such that \(\mu_ 1\geq \mu_ 2\geq...\geq \mu_ i\geq \mu_{n+1}\geq \mu_{i+1}\geq...\geq \mu_ n\) and \(\lambda_ n- \lambda_{n+1}\geq n-i\). Given a shape \(\gamma\) in \(J_ n\) (of weight r), the Weyl module \(K_{\gamma}(F)\) and the Schur module \(L_{\gamma}(F)\) are defined as A-modules. The authors construct resolutions D(\(\gamma\),F), \(\Lambda\) (\(\gamma\),F) for \(K_{\gamma}(F)\), \(L_{\gamma}(F)\), respectively over A. The terms of the complex D(\(\gamma\),F) are sums of tensor products of divided powers and those of \(\Lambda\) (\(\gamma\),F) are sums of tensor products of exterior powers of F. They then study extensions of Weyl modules and Schur modules. The groups \(Ext^ 1_ A(K_{\lambda}(F),K_{\mu}(F))\) are discussed for certain pairs of partitions \(\lambda\) and \(\mu\). Some results of an explicit description of the resolutions that were constructed are sketched in the last section.
reviewed by: Bhama Srinivasan
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 4114884
zbMATH Keywords: free module
zbMATH Keywords: symmetric group
zbMATH Keywords: algebra of \(\Gamma\)-endomorphisms
zbMATH Keywords: Schur algebra
zbMATH Keywords: divided power algebra
zbMATH Keywords: partitions
zbMATH Keywords: Weyl module
zbMATH Keywords: Schur module
zbMATH Keywords: sums of tensor products of divided powers
zbMATH Keywords: exterior powers
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90027-8
OpenAlex ID: W1989660473
cites work: Characteristic-free representation theory of the general linear group
cites work: Schur functors and Schur complexes
cites work: On the modular representations of the general linear and symmetric groups
cites work: On homomorphisms between Weyl modules and Specht modules
cites work: Polynomial representations of \(GL_n\)
cites work: \(\lambda\)-rings and the representation theory of the symmetric group
cites work: Q3947818
description: scientific article; zbMATH DE number 4114884