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Cyclic Sieving for Generalised Non-crossing Partitions Associated with Complex Reflection Groups of Exceptional Type

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Advances in Combinatorics

Abstract

We prove that the generalised non-crossing partitions associated with well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, and by Bessis and Reiner. The computational details are provided in the manuscript “Cyclic sieving for generalised non-crossing partitions associated with complex reflection groups of exceptional type—the details” [arχiv:1001.0030].

Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.

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References

  1. G. E. Andrews, The Theory of Partitions, Encyclopedia of Math. and its Applications, vol. 2, Addison–Wesley, Reading, 1976.

    Google Scholar 

  2. G. E. Andrews, The Friedman–Joichi–Stanton monotonicity conjecture at primes, in: Unusual Applications of Number Theory (M. Nathanson, ed.), DIMACS Ser. Discrete Math. Theor. Comp. Sci., vol. 64, Amer. Math. Soc., Providence, R.I., 2004, pp. 9–15.

    Google Scholar 

  3. D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc., vol. 202, no. 949, Amer. Math. Soc., Providence, R.I., 2009.

    Google Scholar 

  4. D. Armstrong, C. Stump and H. Thomas, A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc. (to appear).

    Google Scholar 

  5. C. A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc. 36 (2004), 294–302.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), 179–196.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. (4) 36 (2003), 647–683.

    Google Scholar 

  8. D. Bessis, Finite complex reflection groups are \(K(\pi,1)\), preprint, ar χ iv:math/0610777.

    Google Scholar 

  9. D. Bessis and R. Corran, Non-crossing partitions of type (e, e, r), Adv. Math. 202 (2006), 1–49.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Bessis and V. Reiner, Cyclic sieving and noncrossing partitions for complex reflection groups, Ann. Comb. 15 (2011), 197–222.

    Article  MathSciNet  MATH  Google Scholar 

  11. T. Brady and C. Watt, \(K(\pi,1)\)’s for Artin groups of finite type, Geom. Dedicata 94 (2002), 225–250.

    Google Scholar 

  12. P. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1981), 171–180.

    Article  MathSciNet  Google Scholar 

  13. S. Fomin and N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Notices 44 (2005), 2709–2757.

    Article  MathSciNet  Google Scholar 

  14. M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE—a system for computing and processing generic character tables for finite groups of Lie type, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.

    Google Scholar 

  15. I. Gordon and S. Griffeth, Catalan numbers for complex reflection groups, Amer. J. Math. (to appear).

    Google Scholar 

  16. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  17. C. Krattenthaler, The F-triangle of the generalised cluster complex, in: Topics in Discrete Mathematics, dedicated to Jarik Nešetřil on the occasion of his 60th birthday (M. Klazar, J. Kratochvil, M. Loebl, J. Matoušek, R. Thomas and P. Valtr, eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126.

    Google Scholar 

  18. C. Krattenthaler, The M-triangle of generalised non-crossing partitions for the types E 7 and E 8, Séminaire Lotharingien Combin. 54 (2006), Article B54l, 34 pages.

    Google Scholar 

  19. C. Krattenthaler, Non-crossing partitions on an annulus, in preparation.

    Google Scholar 

  20. C. Krattenthaler and T. W. Müller, Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions, Trans. Amer. Math. Soc. 362 (2010), 2723–2787.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Krattenthaler and T. W. Müller, Cyclic sieving for generalised non-crossing partitions associated with complex reflection groups of exceptional type—the details, manuscript; ar χ iv:1001.0030.

    Google Scholar 

  22. G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), 333–350.

    Article  MathSciNet  MATH  Google Scholar 

  23. G. I. Lehrer and J. Michel, Invariant theory and eigenspaces for unitary reflection groups, C. R. Math. Acad. Sci. Paris 336 (2003), 795–800.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. I. Lehrer and D. E. Taylor, Unitary reflection groups, Cambridge University Press, Cambridge, 2009.

    MATH  Google Scholar 

  25. G. Malle and J. Michel, Constructing representations of Hecke algebras for complex reflection groups, LMS J. Comput. Math. 13 (2010), 426–450.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. Marin, The cubic Hecke algebra on at most 5 strands, J. Pure Appl. Algebra 216 (2012), 2754–2782.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Michel, The GAP-part of the CHEVIE system, GAP 3-package available for download from http://people.math.jussieu.fr/jmichel/chevie/chevie.html.

  28. P. Orlik and L. Solomon, Unitary reflection groups and cohomology, Invent. Math. 59 (1980), 77–94.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004), 17–50.

    Article  MathSciNet  MATH  Google Scholar 

  30. B. Rhoades, Parking structures: Fuss analogs, preprint, ar χ iv:1205.4293.

    Google Scholar 

  31. V. Ripoll, Orbites d’Hurwitz des factorisations primitives d’un élément de Coxeter, J. Algebra 323 (2010), 1432–1453.

    Article  MathSciNet  MATH  Google Scholar 

  32. B. E. Sagan, The cyclic sieving phenomenon: a survey, Surveys in combinatorics 2011, London Math. Soc. Lecture Note Ser., vol. 392, Cambridge Univ. Press, Cambridge, 2011, pp. 183–233.

    Google Scholar 

  33. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. R. Stembridge, Some hidden relations involving the ten symmetry classes of plane partitions, J. Combin. Theory Ser. A 68 (1994), 372–409.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. R. Stembridge, Canonical bases and self-evacuating tableaux, Duke Math. J. 82 (1996). 585–606,

    Article  MathSciNet  MATH  Google Scholar 

  37. J. R. Stembridge, scoxeter, Maple package for working with root systems and finite Coxeter groups; available at http://www.math.lsa.umich.edu/\~jrs.

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Acknowledgements

The authors thank an anonymous referee for a very careful reading of the original manuscript, and for numerous pertinent suggestions which have helped to considerably improve the original manuscript.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstraß e 15, A-1090, Vienna, Austria

    Christian Krattenthaler

  2. School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London, E1 4NS, UK

    Thomas W. Müller

Authors
  1. Christian Krattenthaler
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  2. Thomas W. Müller
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Corresponding author

Correspondence to Christian Krattenthaler .

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Editors and Affiliations

  1. Wilfrid Laurier University Dept. of Physics and Computer Science, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Wilfrid Laurier University Dept. Physics & Computer Science, Waterloo, Ontario, Canada

    Eugene V. Zima

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Dedicated to the memory of Herb Wilf

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Krattenthaler, C., Müller, T.W. (2013). Cyclic Sieving for Generalised Non-crossing Partitions Associated with Complex Reflection Groups of Exceptional Type. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_12

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