Admissible W-graphs and commuting Cartan matrices

A W-graph is a weighted directed graph that encodes certain actions of a Coxeter group W or the associated Iwahori–Hecke algebra H(W). It is admissible if it is bipartite and has nonnegative integer edge weights that satisfy a simple symmetry condition. Of particular interest are the admissible W-graphs and W×W-graphs that encode the one-sided and two-sided actions of the standard generators on the Kazhdan–Lusztig basis of H(W), as well as the strongly connected components of these graphs—the latter are the so-called Kazhdan–Lusztig cells. Previously, we have posed the problem of classifying the admissible W-cells, and noted the possibility that there may only be finitely many such cells for each finite W. In this paper, we classify the admissible cells for direct products of two dihedral groups. This amounts to classifying pairs of simply-laced (but possibly reducible) Cartan matrices that commute and satisfy a simple parity condition. What we find is that these commuting pairs occur in 6 infinite families along with 11 exceptional pairs whose ranks range from 12 to 32.