2002/03/31 by Jonathan Brundan
Mathematics · #math.RT #math.QA #msc:17B10
published as J. Amer. Math. Soc. 16 (2003), 185-231. · Version 3: lots of corrections and new material explaining relationship between the Kazhdan-Lusztig polynomials and cohomology in finite dimensional case
arxiv created 2002/09/18 · arxiv updated 2009/11/30
The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra \mathfrakgl(m|n) over \C was solved a few years ago by V. Serganova. In this article, we present an entirely different approach. One consequence is a direct and elementary proof of a conjecture made by van der Jeugt and Zhang for the composition multiplicities of Kac modules. This does not seem to follow easily from Serganova's formula, since that involves certain alternating sums. We also compute Ext's between simple modules in the category of finite dimensional representations, and formulate for the first time a conjecture for the characters of the infinite dimensional irreducible representations in the analogue of category \mathcal O for the Lie superalgebra \mathfrakgl(m|n).