## 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.

Original language | English |
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Pages (from-to) | 1419-1452 |

Number of pages | 34 |

Journal | Algebra and Number Theory |

Volume | 9 |

Issue number | 6 |

DOIs | |

State | Published - 22 Sep 2015 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:©2015 Mathematical Sciences Publishers.

### Funding

Funders | Funder number |
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National Science Foundation | |

Directorate for Mathematical and Physical Sciences | 0943832 |

## Keywords

- Character formulas
- Kazhdan–Lusztig polynomials
- Lie superalgebras
- Tame modules