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Representation Theory

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Admissible $ W$-graphs

Author: John R. Stembridge
Journal: Represent. Theory 12 (2008), 346-368
MSC (2000): Primary 20F55, 20C15; Secondary 05E99
Published electronically: October 9, 2008
MathSciNet review: 2448288
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Abstract: Given a Coxeter group $ W$, a $ W$-graph $ \Gamma $ encodes a module $ M_{\Gamma }$ for the associated Iwahori-Hecke algebra $ \mathcal{H}$. The strongly connected components of $ \Gamma $, known as cells, are also $ W$-graphs, and their modules occur as subquotients in a filtration of $ M_{\Gamma }$. Of special interest are the $ W$-graphs and cells arising from the Kazhdan-Lusztig basis for the regular representation of $ \mathcal{H}$. We define a $ W$-graph to be admissible if, like the Kazhdan-Lusztig $ W$-graphs, it is edge-symmetric, bipartite, and has nonnegative integer edge weights. Empirical evidence suggests that for finite $ W$, there are only finitely many admissible $ W$-cells. We provide a combinatorial characterization of admissible $ W$-graphs, and use it to classify the admissible $ W$-cells for various finite $ W$ of low rank. In the rank two case, the nontrivial admissible cells turn out to be $ A$-$ D$-$ E$ Dynkin diagrams.

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Additional Information

John R. Stembridge
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

Received by editor(s): June 8, 2008
Published electronically: October 9, 2008
Additional Notes: This work was supported by NSF Grant DMS–0554278.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.