Non-crossing partitions for classical reflection groups
Discrete Mathematics
Noncrossing Partitions for the Group Dn
SIAM Journal on Discrete Mathematics
Chains in the Noncrossing Partition Lattice
SIAM Journal on Discrete Mathematics
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When W is a finite reflection group, the noncrossing partition lattice $\operatorname{NC}(W)$ of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $\operatorname{NC}(W)$ as a generalized Fuβ---Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $\operatorname{NC}(W)$ as fibers of a Lyashko---Looijenga covering ( $\operatorname{LL}$ ), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map $\operatorname{LL}$ , describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of W.