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

  • 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 case-by-case) expresses the number of multichains of a given length in $NC(W)$ 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 $NC(W)$ as fibers of a "Lyashko-Looijenga 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 $W$.


Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
Section: Proceedings
Published on: January 1, 2011
Imported on: January 31, 2017
Keywords: complex reflection groups,generalized noncrossing partition lattice,generalized Fuss-Catalan numbers,factorizations of a Coxeter element,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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