LyashkoLooijenga morphisms and submaximal factorizations of a Coxeter element
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 FuCatalan 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 LyashkoLooijenga 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.
coxeter element noncrossing partition lattice fucatalan number lyashkolooijenga covering finite coxeter group complex reflection group