Geodesic
Lyashko-Looijenga morphisms and submaximal factorizations of a Coxeter element
Journal of Algebraic Combinatorics, Tome 36 (2012) no. 4, pp. 649-673.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 Fuß-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.
Classification : 23
Keywords: finite Coxeter group, complex reflection group, noncrossing partition lattice, fuß-Catalan number, lyashko-Looijenga covering, Coxeter element 
@article{JAC_2012__36_4_a0,
     author = {Ripoll, Vivien},
     title = {Lyashko-Looijenga morphisms and submaximal factorizations of a {Coxeter} element},
     journal = {Journal of Algebraic Combinatorics},
     pages = {649--673},
     publisher = {mathdoc},
     volume = {36},
     number = {4},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2012__36_4_a0/}
}
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Ripoll, Vivien. Lyashko-Looijenga morphisms and submaximal factorizations of a Coxeter element. Journal of Algebraic Combinatorics, Tome 36 (2012) no. 4, pp. 649-673. http://geodesic.mathdoc.fr/item/JAC_2012__36_4_a0/