Transformation Groups

, Volume 16, Issue 2, pp 555–578 | Cite as

Combinatorics of character formulas for the lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m,n} \right) \)

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Abstract

Let \( \mathfrak{g} \) be the Lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m,n} \right) \). Algorithms for computing the composition factors and multiplicities of Kac modules for \( \mathfrak{g} \) were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph \( \mathcal{G} \) defined using these diagrams. Each vertex of \( \mathcal{G} \) corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If \( \mathcal{E} \) is the subgraph of \( \mathcal{G} \) obtained by deleting all edges of positive weight, then \( \mathcal{E} \) is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.

Keywords

Legal Move Composition Factor Grothendieck Group Character Formula Parabolic Subalgebras 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of WisconsinMilwaukeeUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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