Abstract
We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups.
Similar content being viewed by others
References
Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202, #949 (2009)
Armstrong, D.: Braid groups, clusters, and free probability: an outline from the AIM workshop. Available at www.aimath.org/WWN/braidgroups/braidgroups.pdf (2005)
Athanasiadis, C.A.: On noncrossing and nonnesting partitions for classical reflection groups. Electron. J. Combin. 5, #R42 (1998)
Athanasiadis C.A.: On a refinement of the generalized Catalan numbers for Weyl groups. Trans. Amer. Math. Soc. 357(1), 179–196 (2005)
Athanasiadis C.A., Reiner V.: Noncrossing partitions for the group D n . SIAM J. Discrete Math. 18(2), 397–417 (2004)
Berenstein A., Burman Y.: Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras. Trans. Amer. Math. Soc. 362(1), 229–260 (2010)
Berest Y., Etingof P., Ginzburg V.: Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003(19), 1053–1088 (2003)
Bessis D.: The dual braid monoid. Ann. Sci. Ecole Norm. Sup.(4) 36(5), 647–683 (2003)
Bessis, D.: Finite complex reflection arrangements are K(π, 1). arXiv preprint \({{\tt math.GT/0610777}}\)
Bessis, D.: Garside categories, periodic loops and cyclic sets. arXiv preprint \({{\tt math.GT/0610778}}\)
Brady T., Watt C.: A partial order on the orthogonal group. Comm. Algebra 30(8), 3749–3754 (2002)
Broer, A.: Lectures on decomposition classes. In: Broer, A. (ed.) Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), pp. 39–83. Kluwer Acad. Publ., Dordrecht (1998)
Broué M., Malle G., Rouquier R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)
Chevalley C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)
Eu S.-P., Fu T.-S.: The cyclic sieving phenomenon for faces of generalized cluster complexes. Adv. Appl. Math. 40(3), 350–376 (2008)
Fomin S., Reading N.: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Not. 2005(44), 2709–2757 (2005)
Fomin, S., Reading, N.: Root systems and generalized associahedra. In: Geometric Combinatorics, pp. 63–131. Amer. Math. Soc., Providence, RI (2007)
Fomin S., Zelevinsky A.: Y-systems and generalized associahedra. Ann. of Math. (2) 158(3), 977–1018 (2003)
Fürlinger J., Hofbauer J.: q-Catalan numbers. J. Combin. Theory Ser. A 40(2), 248–264 (1985)
Griffeth, S.: Finite dimensional modules for rational Cherednik algebras. arXiv:math.RT/0612733.
Gordon I.: On the quotient ring by diagonal invariants. Invent. Math. 153(3), 503–518 (2003)
Haiman M.D.: Conjectures on the quotient ring by diagonal invariants. J. Algebraic Combin. 3(1), 17–76 (1994)
Kreweras G.: Sur les partitions non croisées d’un cycle. Discrete Math. 1, 333–350 (1972)
Lehrer G.I., Michel J.: Invariant theory and eigenspaces for unitary reflection groups. C. R. Acad. Sci. Paris, Ser. I 336(10), 795–800 (2003)
Lehrer G.I., Springer T.A.: Reflection subquotients of unitary reflection groups. Canad. J. Math. 51(6), 1175–1193 (1999)
Orlik P., Solomon L.: Unitary reflection groups and cohomology. Invent. Math. 59(1), 77–94 (1980)
Panyushev D.I.: On orbits of antichains of positive roots. European J. Combin. 30(2), 586–594 (2009)
Reiner V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1-3), 195–222 (1997)
Reiner V., Stanton D., White D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108(1), 17–50 (2004)
Shephard G.C., Todd J.A.: Finite unitary reflection groups. Canad. J. Math. 6, 274–304 (1954)
Solomon L.: Invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)
Sommers E.N.: B-stable ideals in the nilradical of a Borel subalgebra. Canad. Math. Bull. 48(3), 460–472 (2005)
Springer T.A.: Regular elements of finite reflection groups. Invent. Math. 25, 159–198 (1974)
White, D.: personal communication. (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Second author supported by NSF grant DMS–9877047.
Rights and permissions
About this article
Cite this article
Bessis, D., Reiner, V. Cyclic Sieving of Noncrossing Partitions for Complex Reflection Groups. Ann. Comb. 15, 197–222 (2011). https://doi.org/10.1007/s00026-011-0090-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-011-0090-9