Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice 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 as a gen...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2012-12, Vol.36 (4), p.649-673
Main Author: Ripoll, Vivien
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary:When W is a finite reflection group, the noncrossing partition lattice 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 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 as fibers of a Lyashko–Looijenga covering ( ), constructed from the geometry of the discriminant hypersurface of W . We study algebraically the map , 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 .
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-012-0354-4