The M-Triangle of Generalised Non-Crossing Partitions for the Types E7 and E8

Christian Krattenthaler

Séminaire lotharingien de combinatoire, Tome 54 (2006-2007)

Résumé

The M-triangle of a ranked locally finite poset P is the generating function \sum _{u,w \in P} \mu(u,w) x^{rk u}y^{rk 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 D_n this had been accomplished in the earlier paper ``The F-triangle of the generalised cluster complex" [in: "Topics in Discrete Mathematics," M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas and P. Valtr, eds., Springer-Verlag, Berlin, New York, 2006, pp. 93-126]. 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 D_n. Moreover, we prove a reciprocity result for this M-triangle, again with the possible exception of type D_n. Our results are based on the calculation of certain decomposition numbers for the reflection groups of types E₇ and E₈, which carry in fact finer information than the M-triangle does. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. As an aside, we show that there is a closed form product formula for the type A_n decomposition numbers, leaving the problem of computing the type B_n and type D_n decomposition numbers open.