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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.