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

Decomposition Numbers for Finite Coxeter Groups and Generalised Non-crossing Partitions

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 = σ1σ2 · · ·σd of a Coxeter element c of W , such that σi 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 t...

Non-crossing partitions for arbitrary Coxeter groups

Eriksson's numbers game and finite Coxeter groups

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many author...

Non-crossing partitions for classical reflection groups

We introduce analogues of the lattice of non-crossing set partitions for the classical reeection groups of type B and D. The type B analogues ((rst considered by Montenegro in a diierent guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain dec...

Non-crossing Partitions, Non-nesting Partitions and Coxeter Sortable Elements in Types a and B

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic reflection groups. In particular, we explore Dyck paths of type B together with an area statistic and a major index. Then, we construct bijections between non-nesting and reverse non-crossing partitions for types A and B. These bijections simultaneously send the area statistic and the major index...