Character formula for finite-dimensional simple representations of the Lie superalgebra \(\mathfrak{gl}(2,2)\)
type: Item
instance of: scholarly article
title: Character formula for finite-dimensional simple representations of the Lie superalgebra \(\mathfrak{gl}(2,2)\)
zbMATH Open document ID: 1223.17012
author: François Drouot
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2010-05-26T00:00:00Z
review text: The purpose of this paper is to find a decomposition of finite-dimensional simple \(\mathfrak{gl}(2,2)\)-modules as a direct sum of simple \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-modules and to use this decomposition in order to find a combinatorial character formula for simple \(\mathfrak{gl}(2,2)\)-modules of finite dimension. A simple \(\mathfrak{gl}(2,2)\)-module is, by restriction, a \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-module and so, since \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\) is a reductive (complex) Lie algebra, it is completely reducible with respect to this Lie algebra. The Cartan subalgebra of diagonal matrices in \(\mathfrak{gl}(2,2)\) is also a Cartan subalgebra of \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\); hence, a combinatorial character formula may be obtained from a \(\mathfrak{gl}(2,2)\)-module decomposition. The simple \(\mathfrak{gl}(2,2)\)-module \(L_\lambda\) of highest weight \(\lambda\) is the unique simple quotient of the Kac module \(K_\lambda\) [\textit{V. G. Kac}, Differ. Geom. Methods Math. Phys. II, Proc., Bonn 1977, Lect. Notes Math. 676, 597--626 (1978; Zbl 0388.17002)] and is of finite dimension if and only if \(\lambda\) is dominant. When \(\lambda\) is dominant, the author finds a decomposition of \(L_\lambda\) as a direct sum of simple \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-modules via the \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-decomposition of \(K_\lambda\). He exploits this decomposition to show that, if \(L_\lambda\) is not of dimension 1, then the conjecture of \textit{J. Bernstein} and \textit{D. Leites} [C. R. Acad. Bulg. Sci. 33, 1049--1051 (1980; Zbl 0457.17002)] for the character of \(L_\lambda\) is verified.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 5712614
zbMATH Keywords: Lie superalgebra
zbMATH Keywords: simple module
zbMATH Keywords: Cartan subalgebra
zbMATH Keywords: highest weight
zbMATH Keywords: dominant weight
zbMATH Keywords: Kac module
zbMATH Keywords: combinatorial character formula
MaRDI profile type: Publication
full work available at URL: j.crma.2010.04.012
OpenAlex ID: W2061357891
cites work: Q3905428
cites work: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔤𝔩(𝔪|𝔫)
cites work: Q3589877
cites work: Characters of typical representations of classical lie superalgebras
cites work: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \({\mathfrak {gl}}(m\mid n)\)
cites work: Character and dimension formulae for general linear superalgebra
DOI: 10.1016/J.CRMA.2010.04.012
Recommended article: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \({\mathfrak {gl}}(m\mid n)\)
Recommended article: Q3535209
Recommended article: Truncated symmetric powers and modular representations of GLn
Recommended article: Unique decomposition of tensor products of irreducible representations of simple algebraic groups
Recommended article: The Kac-Wakimoto character formula for the general linear Lie superalgebra
Recommended article: Q3744338
Recommended article: Finite-dimensional representations of the Lie superalgebra gl(2/2) in a gl(2)⊕gl(2) basis. I. Typical representations
Recommended article: A superdimension formula for \(\mathfrak{gl}(m|n)\)-modules
Recommended article: Gruson-Serganova character formulas and the Duflo-Serganova cohomology functor
Recommended article: Representations of classical Lie superalgebras of type I
description: scientific article; zbMATH DE number 5712614