Filtrations and positive characteristic Howe duality
type: Item
instance of: scholarly article
title: Filtrations and positive characteristic Howe duality
zbMATH Open document ID: 0986.20044
author: George McNinch
published in: Mathematische Zeitschrift
publication date: 2002-02-25T00:00:00Z
review text: Let \(k\) be a field and \(A\) a \(k\)-algebra, finite dimensional over \(k\). Index the simple left \(A\)-modules by a (necessarily finite) set \(\Omega\). For each \(\lambda\in\Omega\), let \(L(\lambda)\), \(P(\lambda)\), \(Q(\lambda)\) denote respectively the simple module corresponding to \(\lambda\), its projective cover, and its injective hull. Assume given a partial ordering \(\leq\) on \(\Omega\). For \(\lambda\in\Omega\), one defines the standard module, or Weyl module, \(\Delta(\lambda)\) as the quotient of \(P(\lambda)\) by the sum of all images \(P(\mu)\to P(\lambda)\) for all \(\mu\not\leq\lambda\) in \(\Omega\). Dually one defines the co-standard module \(\nabla(\lambda)\) as the intersection of all kernels of maps \(Q(\lambda)\to Q(\mu)\) for all \(\mu\not\leq\lambda\) in \(\Omega\). The data \((A,\Omega,\leq)\) is said to be quasi-hereditary provided that \(\leq\) is equivalent to a total ordering on \(\Omega\), the regular module \(_AA\) has a \(\Delta\)-filtration and for each \(\lambda\in\Omega\), \(L(\lambda)\) appears as a composition factor in \(\Delta(\lambda)\) precisely once. Quasi-hereditary algebras were originally introduced by Cline, Parshall and Scott in order to describe highest weight categories arising in the representation theory of complex Lie algebras and algebraic groups; their formulation of the definition was slightly different than, but equivalent to, the above one. The formulation here follows from \textit{C. M. Ringel} [Math. Z. 208, No. 2, 209-224 (1991; Zbl 0725.16011)] (see also [Lond. Math. Soc. Lect. Note Ser. 168, 200-224 (1992; Zbl 0793.16006)] by \textit{C. M. Ringel} and \textit{V. Dlab}). The category of finitely generated \(A\)-modules will be denoted by \(A\)-Mod. A module \(M\) in \(A\)-Mod is said to have a \(\Delta\)-filtration provided there is a sequence of submodules \(\{0\}=M_0\subset M_1\subset\cdots\subset M_r=M\) with \(M_i/M_{i-1}\simeq\Delta(\lambda_i)\) for some \(\lambda_i\in\Omega\) and for each \(…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1559188
zbMATH Keywords: finite reductive groups
zbMATH Keywords: tilting modules
zbMATH Keywords: Jantzen filtrations
zbMATH Keywords: Howe duality
zbMATH Keywords: simple modules
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: equivalences of categories
zbMATH Keywords: Andersen filtrations
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: s002090000157
OpenAlex ID: W1978781621
DOI: 10.1007/S002090000157
description: scientific article; zbMATH DE number 1559188