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Home > Vol 36, No 4 (2012)

Lyashko-Looijenga morphisms and submaximal factorizations of a Coxeter element

Vivien Ripoll

LaCIM, UQÀM, CP 8888, Succ. Centre-ville Montréal, Montréal, QC, H3C 3P8, Canada

DOI: 10.1007/s10801-012-0354-4

Abstract

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.

Pages: 649–673

Keywords: finite Coxeter group; complex reflection group; noncrossing partition lattice; fuß-Catalan number; lyashko-Looijenga covering; Coxeter element

Full Text: PDF

References

Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. $202(949)$, 159 (2009). doi:10.1090/S0065-9266-09-00565-1. Athanasiadis, C.A., Reiner, V.: Noncrossing partitions for the group D $_{ n }$. SIAM J. Discrete Math. $18(2)$, 397-417 (2004). doi:10.1137/S0895480103432192 (electronic) CrossRef Bessis, D.: The dual braid monoid. Ann. Sci. Éc. Norm. Super. $36(5)$, 647-683 (2003). doi:10.1016/j.ansens.2003.01.001 Bessis, D.: Finite complex reflection arrangements are $K(π,1) (2007)$. http://arxiv.org/abs/math/0610777. Preprint arXiv:math/0610777v3 Bessis, D., Corran, R.: Non-crossing partitions of type (e,e,r). Adv. Math. $202(1)$, 1-49 (2006). doi:10.1016/j.aim.2005.03.004 CrossRef Brady, T., Watt, C.: $K(π,1)$'s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I, Haifa, 2000, vol. 94, pp. 225-250 (2002). doi:10.1023/A:1020902610809 Chapoton, F.: Enumerative properties of generalized associahedra. Sémin. Lothar. Comb. 51, Art. B51b (2004) (electronic) Deligne, P.: (1974). Letter to E. Looijenga (9/3/1974) Fomin, S., Reading, N.: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Not. $2005(44)$, 2709-2757 (2005) CrossRef Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. $154(1)$, 63-121 (2003). doi:10.1007/s00222-003-0302-y CrossRef Kane, R.: Reflection Groups and Invariant Theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol.
5. Springer, New York (2001) Krattenthaler, C., Müller, T.W.: Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions. Trans. Am. Math. Soc. 362, 2732-2787 (2010) CrossRef Kreweras, G.: Sur les partitions non croisées d'un cycle. Discrete Math. $1(4)$, 333-350 (1972) CrossRef Lehrer, G.I., Taylor, D.E.: Unitary reflection groups. In: Australian Mathematical Society Lecture Series, vol.
20. Cambridge University Press, Cambridge (2009) Looijenga, E.: The complement of the bifurcation variety of a simple singularity. Invent. Math. 23, 105-116 (1974) CrossRef Picantin, M.: Explicit presentations for the dual braid monoids. C. R. Math. Acad. Sci. Paris $334(10)$, 843-848 (2002) CrossRef Reading, N.: Chains in the noncrossing partition lattice. SIAM J. Discrete Math. $22(3)$, 875-886 (2008). doi:10.1137/07069777X CrossRef Reiner, V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1-3), 195-222 (1997). doi:10.1016/S0012-$365X(96)00365$-2 CrossRef Ripoll, V.: Discriminants and Jacobians of virtual reflection groups (2010). http://arxiv.org/abs/1001.4470. Preprint arXiv:1001.4470 Ripoll, V.: Groupes de réflexion, géométrie du discriminant et partitions non-croisées. Ph.D. thesis, Université Paris Diderot-Paris 7 (2010). http://arxiv.org/abs/1010.4349. arXiv:1010.4349 Ripoll, V.: Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter. J. Algebra $323(5)$, 1432-1453 (2010). doi:10.1016/j.jalgebra.2009.12.010 CrossRef Saito, K.: Polyhedra dual to the Weyl chamber decomposition: a précis. Publ. Res. Inst. Math. Sci. $40(4)$, 1337-1384 (2004). http://projecteuclid.org/getRecord?id=euclid.prims/1145475449 CrossRef Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274-304 (1954) CrossRef Springer, T.A.: Regular elements of finite reflection groups. Invent. Math. 25, 159-198 (1974) CrossRef Stanley, R.P.: Enumerative Combinatorics, vol.
1. Cambridge Studies in Advanced Mathematics, vol.
49. Cambridge University Press, Cambridge (1997) CrossRef

	EMIS Electronic Library of Mathematics (ELibM) The Open Access Repository of Mathematics
	JOURNAL OF ALGEBRAIC COMBINATORICS
	Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
	Home > Vol 36, No 4 (2012) Lyashko-Looijenga morphisms and submaximal factorizations of a Coxeter element Vivien Ripoll LaCIM, UQÀM, CP 8888, Succ. Centre-ville Montréal, Montréal, QC, H3C 3P8, Canada DOI: 10.1007/s10801-012-0354-4 Abstract 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. Pages: 649–673 Keywords: finite Coxeter group; complex reflection group; noncrossing partition lattice; fuß-Catalan number; lyashko-Looijenga covering; Coxeter element Full Text: PDF References Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. $202(949)$, 159 (2009). doi:10.1090/S0065-9266-09-00565-1. Athanasiadis, C.A., Reiner, V.: Noncrossing partitions for the group D $_{ n }$. SIAM J. Discrete Math. $18(2)$, 397-417 (2004). doi:10.1137/S0895480103432192 (electronic) CrossRef Bessis, D.: The dual braid monoid. Ann. Sci. Éc. Norm. Super. $36(5)$, 647-683 (2003). doi:10.1016/j.ansens.2003.01.001 Bessis, D.: Finite complex reflection arrangements are $K(π,1) (2007)$. http://arxiv.org/abs/math/0610777. Preprint arXiv:math/0610777v3 Bessis, D., Corran, R.: Non-crossing partitions of type (e,e,r). Adv. Math. $202(1)$, 1-49 (2006). doi:10.1016/j.aim.2005.03.004 CrossRef Brady, T., Watt, C.: $K(π,1)$'s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I, Haifa, 2000, vol. 94, pp. 225-250 (2002). doi:10.1023/A:1020902610809 Chapoton, F.: Enumerative properties of generalized associahedra. Sémin. Lothar. Comb. 51, Art. B51b (2004) (electronic) Deligne, P.: (1974). Letter to E. Looijenga (9/3/1974) Fomin, S., Reading, N.: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Not. $2005(44)$, 2709-2757 (2005) CrossRef Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. $154(1)$, 63-121 (2003). doi:10.1007/s00222-003-0302-y CrossRef Kane, R.: Reflection Groups and Invariant Theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 5. Springer, New York (2001) Krattenthaler, C., Müller, T.W.: Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions. Trans. Am. Math. Soc. 362, 2732-2787 (2010) CrossRef Kreweras, G.: Sur les partitions non croisées d'un cycle. Discrete Math. $1(4)$, 333-350 (1972) CrossRef Lehrer, G.I., Taylor, D.E.: Unitary reflection groups. In: Australian Mathematical Society Lecture Series, vol. 20. Cambridge University Press, Cambridge (2009) Looijenga, E.: The complement of the bifurcation variety of a simple singularity. Invent. Math. 23, 105-116 (1974) CrossRef Picantin, M.: Explicit presentations for the dual braid monoids. C. R. Math. Acad. Sci. Paris $334(10)$, 843-848 (2002) CrossRef Reading, N.: Chains in the noncrossing partition lattice. SIAM J. Discrete Math. $22(3)$, 875-886 (2008). doi:10.1137/07069777X CrossRef Reiner, V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1-3), 195-222 (1997). doi:10.1016/S0012-$365X(96)00365$-2 CrossRef Ripoll, V.: Discriminants and Jacobians of virtual reflection groups (2010). http://arxiv.org/abs/1001.4470. Preprint arXiv:1001.4470 Ripoll, V.: Groupes de réflexion, géométrie du discriminant et partitions non-croisées. Ph.D. thesis, Université Paris Diderot-Paris 7 (2010). http://arxiv.org/abs/1010.4349. arXiv:1010.4349 Ripoll, V.: Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter. J. Algebra $323(5)$, 1432-1453 (2010). doi:10.1016/j.jalgebra.2009.12.010 CrossRef Saito, K.: Polyhedra dual to the Weyl chamber decomposition: a précis. Publ. Res. Inst. Math. Sci. $40(4)$, 1337-1384 (2004). http://projecteuclid.org/getRecord?id=euclid.prims/1145475449 CrossRef Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274-304 (1954) CrossRef Springer, T.A.: Regular elements of finite reflection groups. Invent. Math. 25, 159-198 (1974) CrossRef Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997) CrossRef © 1992–2009 Journal of Algebraic Combinatorics © 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition

JOURNAL OF ALGEBRAIC COMBINATORICS

Lyashko-Looijenga morphisms and submaximal factorizations of a Coxeter element

Abstract

References

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