The M-triangle of generalised non-crossing partitions for the types E7 and E8

(34 pages)

Abstract. The M-triangle of a ranked locally finite poset P is the generating function \sum _{u,w \in P} \mu(u,w) xrk u yrk w, where \mu(.,.) is the Möbius function of P. We compute the M-triangle of Armstrong's poset of m-divisible non-crossing partitions for the root systems of type E7 and E8. For the other types except Dn this had been accomplished in the earlier paper The F-triangle of the generalised cluster complex." Altogether, this almost settles Armstrong's F=M Conjecture predicting a surprising relation between the M-triangle of the m-divisible partitions poset and the F-triangle (a certain refined face count) of the generalised cluster complex of Fomin and Reading, the only gap remaining in type Dn. Moreover, we prove a reciprocity result for this M-triangle, again with the possible exception of type Dn. Our results are based on the calculation of certain decomposition numbers for the reflexion groups of types E7 and E8, which carry in fact finer information than does the M-triangle. The decomposition numbers for the other exceptional reflexion groups had been computed in the earlier paper. We present a conjectured formula for the type An decomposition numbers.
Here are the Mathematica inputs for the decomposition number computations reported in the paper.

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