Modular representations of the supergroup \(Q(n)\). I
type: Item
instance of: scholarly article
title: Modular representations of the supergroup \(Q(n)\). I
zbMATH Open document ID: 1027.17004
DOI: 10.1016/S0021-8693(02)00620-8
author: Jonathan Brundan
author: Alexander S. Kleshchev
published in: Journal of Algebra
publication date: 2003-04-28T00:00:00Z
review text: The authors study the representation theory of the algebraic supergroup \(Q(n)\) in positive characteristic \(p\). The study starts with a construction of a special superalgebra, \(\text{Dist}(G)\), of distributions on \(G\). It is shown that there is an explicit equivalence between the category of representations of \(G\) and the category of ``integrable'' \(\text{Dist}(G)\)-supermodules. Further, it is shown that the irreducible representations of \(G\) can be characterized in terms of highest weights. They turn out to be parameterized by the set \[ X_p^+(n)=\left\{ (\lambda_1,\dots,\lambda_n)\in\mathbb{Z}^n : \lambda_1\geq\dots\geq \lambda_n \text{ with } \lambda_i=\lambda_{i+1} \text{ only if } p|\lambda_i \right\}. \] For \(\lambda\in X_p^+(n)\) the corresponding irreducible representation \(L(\lambda)\) is the simple socle of the induced module \(H^0(\lambda)=\text{ind}_B^G\mathfrak{u}(\lambda)\), where \(B\) is a Borel subgroup of \(G\) and \(\mathfrak{u}(\lambda)\) is a certain irreducible representation of \(B\), whose dimension is a power of \(2\). It is shown that, unlike the classical situation of algebraic groups, the self-extensions between simple modules are possible, arising from the fact that representations of the diagonal subgroup, which plays the role of the maximal torus, are not completely reducible. The authors also prove a linkage principle involving the notion of residue content of a weight \(\lambda\in X_p^+(n)\). In particular, it follows that for any two irreducible representations whose parameters have different contents, the first extension space between these two representations is zero. Further, the authors prove an analogue of the Steinberg tensor product theorem, showing that any irreducible representation of \(Q(n)\) decomposes into a tensor product of a restricted irreducible representation and one obtained by a Frobenius twist. The restricted irreducible representations of \(Q(n)\) are exactly those which remain irreducible over …
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20G99
zbMATH DE Number: 1901817
zbMATH Keywords: supergroup
zbMATH Keywords: representation
zbMATH Keywords: category
zbMATH Keywords: highest weight
zbMATH Keywords: extension
zbMATH Keywords: irreducible representation
MaRDI profile type: Publication
cites work: Q5594655
cites work: Projective representations of symmetric groups via Sergeev duality
cites work: On the modular representations of the general linear and symmetric groups
cites work: On the tensor product theorem for algebraic groups
cites work: Q5602064
cites work: Hopf Complements and Injective Comodules for Algebraic Groups
cites work: On Schur algebras and related algebras. I
cites work: On Schur algebras and related algebras. II
cites work: Standard homological properties for quantum \(GL_ n\)
cites work: Cohomology of finite group schemes over a field
cites work: Q3752572
cites work: Q4378365
cites work: Lie superalgebras
cites work: Q5596902
cites work: q-Deformed Fock spaces and modular representations of spin symmetric groups
cites work: INTRODUCTION TO THE THEORY OF SUPERMANIFOLDS
cites work: Q4225020
cites work: Characters of typical irreducible finite-dimensional \({\mathfrak q}(n)\)- modules
cites work: Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules
cites work: Characters of irreducible \(G\)-modules and cohomology of \(G/P\) for the Lie supergroup \(G=Q(N)\)
cites work: Character of finite dimensional irreducible \(q(n)\)-modules
cites work: The theory of Lie superalgebras. An introduction
cites work: THE TENSOR ALGEBRA OF THE IDENTITY REPRESENTATION AS A MODULE OVER THE LIE SUPERALGEBRAS $ \mathfrak{Gl}(n,\,m)$ AND $ Q(n)$
cites work: The centre of enveloping algebra for Lie superalgebra Q(n,\({\mathbb{C}})\)
cites work: Lectures on Chevalley Groups
cites work: Representations of Algebraic Groups
full work available at URL: s0021-8693(02)00620-8
OpenAlex ID: W2006343342
description: scientific article; zbMATH DE number 1901817