Semistandard filtrations in highest weight categories.
type: Item
instance of: scholarly article
title: Semistandard filtrations in highest weight categories.
zbMATH Open document ID: 1182.20044
published in: Michigan Mathematical Journal
publication date: 2009-06-17T00:00:00Z
review text: The author defines the new concept of a semistandard filtration in a highest weight category. Basic properties of semistandard filtrations are presented. For any semistandard filtration of a module \(M\), a semistandard multiplicity associated to a standard module in \(M\) is defined, and proved that it is independent on the choice of the semistandard filtration. The special case of semistandard filtrations of maximal submodules of standard modules is considered, and a characterization of the semistandard multiplicities in terms of the dimensions of \(\text{Ext}^1\) groups between irreducible modules is given. Applications to some general inequalities on the behavior of \(\text{Ext}^1\) in the presence of a suitable exact functor are given. Specific results about \(\text{Ext}^1_G(L,L')\) are proved for irreducible modules \(L,L'\) of a semisimple algebraic group \(G\). These are related to known issues for \(\text{Ext}^n\) and parity conditions involving standard and irreducible modules with singular high weights in the presence of the Lusztig conjecture for representations of \(G\) in positive characteristics.
reviewed by: Constantin Năstăsescu
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 5566087
zbMATH Keywords: semistandard filtrations
zbMATH Keywords: highest weight categories
zbMATH Keywords: standard modules
zbMATH Keywords: semisimple algebraic groups
zbMATH Keywords: irreducible representations
author: Leonard L. Scott
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1242071697
OpenAlex ID: W2125938161
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DOI: 10.1307/MMJ/1242071697
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description: scientific article; zbMATH DE number 5566087