Characteristic-free representation theory of the general linear group. II: Homological considerations
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.
sums of tensor products of divided powers
scientific article; zbMATH DE number 4114884