Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Abstract

Given a finite irreducible Coxeter group W W , a positive integer d d , and types T 1 , T 2 , … , T d T_1,T_2,\dots ,T_d (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ 1 σ 2 ⋯ σ d c=\sigma _1\sigma _2\cdots \sigma _d of a Coxeter element c c of W W , such that σ i \sigma _i is a Coxeter element in a subgroup of type T i T_i in W W , i = 1 , 2 , … , d i=1,2,\dots ,d , and such that the factorisation is “minimal” in the sense that the sum of the ranks of the T i T_i ’s, i = 1 , 2 , … , d i=1,2,\dots ,d , equals the rank of W W . For the exceptional types, these decomposition numbers have been computed by the first author in [“Topics in Discrete Mathematics,” M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [Séminaire Lotharingien Combin. 54 (2006), Article B54l]. The type A n A_n decomposition numbers have been computed by Goulden and Jackson in [Europ. J. Combin. 13 (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type B n B_n decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our formula for the type D n D_n decomposition numbers is new. These results are then used to determine, for a fixed positive integer l l and fixed integers r 1 ≤ r 2 ≤ ⋯ ≤ r l r_1\le r_2\le \dots \le r_l , the number of multi-chains π 1 ≤ π 2 ≤ ⋯ ≤ π l \pi _1\le \pi _2\le \dots \le \pi _l in Armstrong’s generalised non-crossing partitions poset, where the poset rank of π i \pi _i equals r i r_i and where the “block structure” of π 1 \pi _1 is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type D n D_n generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s F = M F=M Conjecture in type D n D_n , thus completing a computational proof of the F = M F=M Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.