Vol. 9, No. 6, 2015

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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
Abstract

We prove the Kac–Wakimoto character formula for the general linear Lie superalgebra gl(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