#### Vol. 9, No. 6, 2015

 Download this article For screen For printing
 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
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 $\mathfrak{g}\mathfrak{l}\left(m|n\right)$, 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