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

الأصل · EN

Given a finite irreducible Coxeter group W, a positive integer d, and types T₁,T₂,...,Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c=₁₂ cdots of a Coxeter element c of W, such that ᵢ is a Coxeter element in a subgroup of type Tᵢ in W, i=1,2,...,d, and such that the factorisation is "minimal" in the sense that the sum of the ranks of the Tᵢ's, i=1,2,...,d, equals the rank of W. For the exceptional types, these decomposition numbers have been computed by the first author. The type Aₙ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type Bₙ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type Dₙ decomposition numbers is new. These results are then used to determine, for a fixed positive integer l and fixed integers r₁≤ r₂≤...≤ rₗ, the number of multi-chains π₁≤ π₂≤...≤ πₗ in Armstrong's generalised non-crossing partitions poset, where the poset rank of πᵢ equals rᵢ, and where the "block structure" of π₁ 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 Dₙ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong's F=M Conjecture in type Dₙ.