The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$

Abstract

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 M\"obius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$ 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 $D_n$. 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 $E_7$ and $E_8$, which carry in fact finer information than does the $M$-triangle. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. We present a conjectured formula for the type $A_n$ decomposition numbers.