Skip to main content
SpringerLink
Log in

Weyl Group \({\varvec{q}}\)-Kreweras Numbers and Cyclic Sieving

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In fact, Bessis’s work in [6] deals not just with Weyl groups, but all finite real reflection groups, and his later work in [7] deals more generally with the class of well-generated complex reflection groups. See work of Gordon and Griffeth [21] for definitions of the Catalan and q-Catalan numbers that apply to all complex reflection groups.

  2. Examination of the tables in Sect. 3.6 shows that for e in an ill-behaved nilpotent orbit, for certain \(\phi \ne 1\) one still has a factorization of \(f_{e,\phi }(m;q)\) as in the theorem, but with \(-g_\phi (m;q)\) in \({{\mathbb {N}}}[q]\). Also, for such e, property (iii) fails even if \(\phi =1\). Instead, the value \(r-c\) is the multiplicity of V in the A(e)-invariants in \(H^*({\mathcal {B}}_e)\).

  3. B. Rhoades, personal communication, 2016.

References

  1. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202(949), (2009)

  2. Armstrong, D., Rhoades, B., Williams, N.: Rational associahedra and noncrossing partitions. Electron. J. Combin. 20(3), #R54 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Athanasiadis, C.A.: On noncrossing and nonnesting partitions for classical reflection groups. Electron. J. Combin. 5, #R42 (1998)

  4. Athanasiadis, C.A., Reiner, V.: Noncrossing partitions for the group \(D_n\). SIAM J. Discrete Math. 18(2), 397–417 (2004)

    Article  MathSciNet  Google Scholar 

  5. Berest, Y., Etingof, P., Ginzburg, V.: Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003(19), 1053–1088 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bessis, D.: The dual braid monoid. Ann. Sci. École Norm. Sup. (4) 36(5), 647–683 (2003)

    Article  MathSciNet  Google Scholar 

  7. Bessis, D.: Finite complex reflection arrangements are \(K(\pi ,1)\). Ann. of Math. (2) 181(3), 809–904 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bessis, D., Reiner, V.: Cyclic sieving of noncrossing partitions for complex reflection groups. Ann. Combin. 15(2), 197–222 (2011)

    Article  MathSciNet  Google Scholar 

  9. Bodnar, M., Rhoades, B.: Cyclic sieving and rational Catalan theory. Electron. J. Combin. 23(2), #P2.4 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Borho, W., MacPherson, R.: Partial resolutions of nilpotent varieties. In: Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque, Vols. 101–102. pp. 23–74. Soc. Math. France, Paris (1983)

  11. Brändén, P.: \(q\)-Narayana numbers and the flag h-vector of \(J(2\times n)\). Discrete Math. 281(1–3), 67–81 (2004)

    Article  MathSciNet  Google Scholar 

  12. Broer, A.: Lectures on decomposition classes. In: Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 514, pp. 39–83. Kluwer Acad. Publ., Dordrecht (1998)

  13. Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Carter, R.W.: Finite Groups of Lie Type. John Wiley & Sons, Inc., New York (1985)

    MATH  Google Scholar 

  15. Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)

    Google Scholar 

  16. Crabb, M.C.: Counting nilpotent endomorphisms. Finite Fields Appl. 12(1), 151–154 (2006)

    Article  MathSciNet  Google Scholar 

  17. Douglass, J.M.: The adjoint representation of a reductive group and hyperplane arrangements. Represent. Theory 3, 444–456 (1999)

    Article  MathSciNet  Google Scholar 

  18. Edelman, P.H.: Chain enumeration and noncrossing partitions. Discrete Math. 31(2), 171–180 (1980)

    Article  MathSciNet  Google Scholar 

  19. Fürlinger, J., Hofbauer, J.: \(q\)-Catalan numbers. J. Combin. Theory Ser. A 40(2), 248–264 (1985)

    Article  MathSciNet  Google Scholar 

  20. Gordon, I.: On the quotient ring by diagonal invariants. Invent. Math. 153(3), 503–518 (2003)

    Article  MathSciNet  Google Scholar 

  21. Gordon, I., Griffeth, S.: Catalan numbers for complex reflection groups. Amer. J. Math. 134(6), 1491–1502 (2012)

    Article  MathSciNet  Google Scholar 

  22. Hotta, R.: On Springer’s representations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 863–876 (1981)

    MathSciNet  MATH  Google Scholar 

  23. Ito, Y., Okada, S.: On the existence of generalized parking spaces for complex reflection groups. arXiv:1508.06846 [math.CO].

  24. Kim, J.S.: Chain enumeration of \(k\)-divisible noncrossing partitions of classical types. J. Combin. Theory Ser. A 118(3), 879–898 (2011)

    Article  MathSciNet  Google Scholar 

  25. Kim, J.S.: Cyclic sieving phenomena on annular noncrossing permutations. Sém. Lothar. Combin. 69, Art. B69b (2012)

  26. Krattenthaler, C., Müller, T.W.: Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions. Trans. Amer. Math. Soc. 362(5), 2723–2787 (2010)

    Article  MathSciNet  Google Scholar 

  27. Krattenthaler, C., Müller, T.W.: Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type–the details. arXiv:1001.0030 [math.CO]

  28. Kreweras, G.: Sur les partitions non croisées d’un cycle. Discrete Math. 1(4), 333–350 (1972)

    Article  MathSciNet  Google Scholar 

  29. Lehrer, G.I., Shoji, T.: On flag varieties, hyperplane complements and Springer representations of Weyl groups. J. Austral. Math. Soc. Ser. A 49(3), 449–485 (1990)

    Article  MathSciNet  Google Scholar 

  30. Lusztig, G.: A note on counting nilpotent matrices of fixed rank. Bull. London Math. Soc 8(1), 77–80 (1976)

    Article  MathSciNet  Google Scholar 

  31. Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. in Math. 42(2), 169–178 (1981)

    Article  MathSciNet  Google Scholar 

  32. Lusztig, G.: Notes on unipotent classes. Asian J. Math. 1(1), 194–207 (1997)

    Article  MathSciNet  Google Scholar 

  33. Orlik, P., Solomon, L.: Coxeter arrangements. In: Orlik, P. (ed.) Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., Vol. 40, pp. 269–291, Amer. Math. Soc., Providence, RI (1983)

  34. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, Vol. 300. Springer-Verlag, Berlin (1992)

  35. Reiner, V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1–3), 195–222 (1997)

    Article  MathSciNet  Google Scholar 

  36. Reiner, V., Stanton, D., White, D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108(1), 17–50 (2004)

    Article  MathSciNet  Google Scholar 

  37. Rhoades, B.: Parking structures: Fuss analogs. J. Algebraic Combin. 40(2), 417–473 (2014)

    Article  MathSciNet  Google Scholar 

  38. SageMath, The Sage Mathematics Software System (Version 4.7.2). The Sage Developers (2011), http://www.sagemath.org.

  39. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Canadian J. Math. 6, 274–304 (1954)

    Article  MathSciNet  Google Scholar 

  40. Shoji, T.: On the Green polynomials of classical groups. Invent. Math. 74(2), 239–267 (1983)

    Article  MathSciNet  Google Scholar 

  41. Solomon, L.: Invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)

    Article  MathSciNet  Google Scholar 

  42. Sommers, E.: A family of affine Weyl group representations. Transform. Groups 2(4), 375–390 (1997)

    Article  MathSciNet  Google Scholar 

  43. Sommers, E.: Exterior powers of the reflection representation in Springer theory. Transform. Groups 16(3), 889–911 (2011)

    Article  MathSciNet  Google Scholar 

  44. Spaltenstein, N.: On the reflection representation in Springer’s theory. Comment. Math. Helv. 66(4), 618–636 (1991)

    Article  MathSciNet  Google Scholar 

  45. Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)

    Article  MathSciNet  Google Scholar 

  46. Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York (2015)

    Book  Google Scholar 

  47. Stanley, R.P.: Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, Vol. 62. Cambridge University Press, Cambridge (1999)

  48. Terao, H.: Arrangements of hyperplanes and their freeness I, II. The Coxeter equality. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 293–320 (1980)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author thanks Jang Soo Kim for helpful conversations.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Victor Reiner

  2. Department of Mathematics and Statistics, University of Massachusetts-Amherst, Amherst, MA, 01003, USA

    Eric Sommers

Authors
  1. Victor Reiner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric Sommers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Reiner.

Additional information

First author supported by NSF Grant DMS-1001933, second author supported by NSA Grant H98230-11-1-0173 and by a National Science Foundation Independent Research and Development plan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reiner, V., Sommers, E. Weyl Group \({\varvec{q}}\)-Kreweras Numbers and Cyclic Sieving. Ann. Comb. 22, 819–874 (2018). https://doi.org/10.1007/s00026-018-0408-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-018-0408-y

Keywords

Mathematics Subject Classification

Search

Navigation