The Kac–Wakimoto character formula for the general linear Lie superalgebra

Chmutov, Michael; Hoyt, Crystal; Reif, Shifra

doi:10.2140/ant.2015.9.1419

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The Kac–Wakimoto character formula for the general linear Lie superalgebra

Michael Chmutov, Crystal Hoyt and Shifra Reif

Vol. 9 (2015), No. 6, 1419–1452

DOI: 10.2140/ant.2015.9.1419

Abstract

We prove the Kac–Wakimoto character formula for the general linear Lie superalgebra $g l (m | n)$ , which was conjectured by Kac and Wakimoto in 1994. This formula specializes to the well-known Kac–Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules.

In our proof, we demonstrate how to use odd reflections to move character formulas between the different sets of simple roots of a Lie superalgebra. As a consequence, we show that KW-modules are precisely Kostant modules, which were studied by Brundan and Stroppel, thus yielding a simple combinatorial defining condition for KW-modules and a classification of these modules.

Keywords

character formulas, Kazhdan–Lusztig polynomials, Lie superalgebras, tame modules

Mathematical Subject Classification 2010

Primary: 17B10

Secondary: 17B20, 22E47

Milestones

Received: 16 June 2014

Revised: 17 February 2015

Accepted: 28 March 2015

Published: 7 September 2015

Authors

Michael Chmutov
Department of Mathematics University of Minnesota Minneapolis, MN 55455 United States

Crystal Hoyt
Department of Mathematics Technion – Israel Institute of Technology 3200003 Haifa Israel

Shifra Reif
Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States

Vol. 9, No. 6, 2015