Vivien Ripoll

Submaximal factorizations of a Coxeter element in complex reflection groups
dmtcs:2955 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2955
Submaximal factorizations of a Coxeter element in complex reflection groupsArticle
Authors: Vivien Ripoll ^{1}
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Vivien Ripoll
1 Laboratoire de combinatoire et d'informatique mathématique [Montréal]
When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$gon. A formula (for which the only known proofs are casebycase) expresses the number of multichains of a given length in $NC(W)$ as a generalized FußCatalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a casefree way, using an interpretation of the chains of $NC(W)$ as fibers of a "LyashkoLooijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$.