Modular representations of the supergroup \(Q(n)\). II.
type: Item
instance of: scholarly article
title: Modular representations of the supergroup \(Q(n)\). II.
zbMATH Open document ID: 1122.20022
author: Jonathan Brundan
published in: Pacific Journal of Mathematics
publication date: 2007-02-13T00:00:00Z
review text: This is the second of two articles investigating the representation theory of the supergroup \(G=Q(n)\) over an algebraically closed field of characteristic \(p>2\). In contrast to the first article [\textit{J. Brundan, A. Kleshchev}, J. Algebra 260, No. 1, 64-98 (2003; Zbl 1027.17004)] the present article is concerned with results that in some way depend on the interpretation of the induction from a Borel subgroup \(B\) in terms of sheaf cohomology on the flag variety \(G/B\). The irreducible representations of \(G\) are parametrized by certain vectors \(\lambda\) with integer coordinates. The corresponding representation \(L(\lambda)\) can be realized as the unique irreducible submodule of the module \(H^0(\lambda)=H^0(G/B,\mathcal L(\mathfrak u(\lambda)))\), where \(\mathfrak u(\lambda)\) is a certain irreducible \(B\)-supermodule. The first main result of the paper states that the Euler characteristic \(E_\lambda\) of the characters of all higher cohomology modules \(H^i(\lambda)\) is equal to Schur's \(P\)-function, scaled by \(\dim\mathfrak u(\lambda)\). The second main result establishes the linkage principle for the Euler characteristics. Namely, it is shown that if one writes \(E_\lambda\) as the linear combination of characters of irreducible representations, then if the representation \(L(\mu)\) appears in this linear combination with a non-zero coefficient, then \(\mu\) and \(\lambda\) have the same residue content. The coefficients above are called the decomposition numbers. At the end of the paper one finds decomposition matrices for the polynomial representations of \(Q(n)\) of degree \(d\leq 12\) in characteristics \(3\) (with \(n\) large).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14L10
zbMATH DE Number: 5125698
zbMATH Keywords: supergroups
zbMATH Keywords: modular representations
zbMATH Keywords: characters
zbMATH Keywords: sheaves
zbMATH Keywords: vector bundles
zbMATH Keywords: Borel subgroups
zbMATH Keywords: sheaf cohomology
zbMATH Keywords: flag varieties
zbMATH Keywords: irreducible representations
zbMATH Keywords: irreducible supermodules
zbMATH Keywords: cohomology modules
zbMATH Keywords: linkage principles
zbMATH Keywords: decomposition numbers
MaRDI profile type: Publication
full work available at URL: pjm.2006.224.65
OpenAlex ID: W1987106144
DOI: 10.2140/PJM.2006.224.65
description: scientific article; zbMATH DE number 5125698