Admissible $W$-graphs

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.