Tilting modules for classical groups and Howe duality in positive characteristic
type: Item
instance of: scholarly article
title: Tilting modules for classical groups and Howe duality in positive characteristic
zbMATH Open document ID: 0857.20018
DOI: 10.1007/BF02587733
author: Yanyan Li
published in: Transformation Groups
publication date: 1997-03-03T00:00:00Z
review text: To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field \(k\), the authors set in correspondence another series of classical groups. Let \(G_1(m)\) denote a group of rank \(m\) from the first series and \(G_2(n)\) a group of rank \(n\) from the second series. For any pair \((G_1(m),G_2(n))\) the authors construct a \(G_1(m)\times G_2(n)\)-module \(M(m,n)\). For \(\text{char }k=0\) the actions of \(G_1(m)\) and \(G_2(n)\) on \(M(m,n)\) form a reductive dual pair in the sense of Howe. It is shown that \(M(m,n)\) is a tilting \(G_1(m)\)- and \(G_2(n)\)-module and that \(\text{End}_{G_1(m)}M(m,n)\) is generated by \(G_2(n)\) and vice versa. The authors consider the category \({\mathcal K}_1(m,n)\) of rational \(G_1(m)\)-modules with highest weights bounded in a certain sense by \(n\) and the category \({\mathcal K}_2(m,n)\) of rational \(G_2(n)\)-modules with highest weights bounded in a certain sense by \(m\). It is shown that there is a bijection of the set of dominant weights of \(G_1(m)\)-modules from \({\mathcal K}_1(m,n)\) to the set of dominant weights of \(G_2(n)\)-modules from \({\mathcal K}_2(m,n)\) such that Ext groups for induced \(G_1(m)\)-modules from \({\mathcal K}_1(m,n)\) are isomorphic to Ext groups for corresponding Weyl modules over \(G_2(n)\). The derived categories \(D^b{\mathcal K}_1(m,n)\) and \(D^b{\mathcal K}_2(m,n)\) appear to be equivalent. These results are applied to finding generators and relations for the algebra of all \(G\)-invariants in \(\wedge^\bullet(V^{n_1}\oplus(V^*)^{n_2})\), where \(G=GL_m\), \(Sp_{2m}\), \(O_m\), and \(V\) is the natural \(G\)-module.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 934930
zbMATH Keywords: tilting modules
zbMATH Keywords: category of rational modules
zbMATH Keywords: induced modules
zbMATH Keywords: algebra of invariants
zbMATH Keywords: series of classical groups
zbMATH Keywords: actions
zbMATH Keywords: reductive dual pairs
zbMATH Keywords: dominant weights
zbMATH Keywords: Ext groups
zbMATH Keywords: Weyl modules
zbMATH Keywords: derived categories
zbMATH Keywords: generators
zbMATH Keywords: relations
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 934930