## Admissible $W$-graphs

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- by John R. Stembridge
- Represent. Theory
**12**(2008), 346-368 - DOI: https://doi.org/10.1090/S1088-4165-08-00336-1
- Published electronically: October 9, 2008
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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.## References

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## Bibliographic Information

**John R. Stembridge**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
- Email: jrs@umich.edu
- Received by editor(s): June 8, 2008
- Published electronically: October 9, 2008
- Additional Notes: This work was supported by NSF Grant DMS–0554278.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**12**(2008), 346-368 - MSC (2000): Primary 20F55, 20C15; Secondary 05E99
- DOI: https://doi.org/10.1090/S1088-4165-08-00336-1
- MathSciNet review: 2448288