Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 130, 40 pages      arXiv:1907.09337      https://doi.org/10.3842/SIGMA.2020.130

Cyclic Sieving for Plane Partitions and Symmetry

Sam Hopkins
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received May 05, 2020, in final form December 06, 2020; Published online December 09, 2020

Abstract
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

Key words: plane partitions; cyclic sieving phenomena; promotion; rowmotion; canonical bases.

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References

  1. Abuzzahab O., Korson M., Chun M.L.M., Meyer S., Cyclic and dihedral sieving phenomenon for plane partitions, REU report, 2005, available at http://www-users.math.umn.edu/ reiner/REU/PlanePartitionReport.pdf.
  2. Alexandersson P., Linusson S., Potka S., The cyclic sieving phenomenon on circular Dyck paths, Electron. J. Combin. 26 (2019), 4.16, 32 pages, arXiv:1903.01327.
  3. Andrews G.E., Plane partitions. II. The equivalence of the Bender-Knuth and MacMahon conjectures, Pacific J. Math. 72 (1977), 283-291.
  4. Andrews G.E., Plane partitions. I. The MacMahon conjecture, in Studies in Foundations and Combinatorics, Adv. in Math. Suppl. Stud., Vol. 1, Academic Press, New York - London, 1978, 131-150.
  5. Armstrong D., Stump C., Thomas H., A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc. 365 (2013), 4121-4151, arXiv:1101.1277.
  6. Barcelo H., Reiner V., Stanton D., Bimahonian distributions, J. Lond. Math. Soc. 77 (2008), 627-646, arXiv:math.CO/0703479.
  7. Bender E.A., Knuth D.E., Enumeration of plane partitions, J. Combinatorial Theory Ser. A 13 (1972), 40-54.
  8. Berenstein A., Zelevinsky A., Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics, Duke Math. J. 82 (1996), 473-502.
  9. Bloom J., Pechenik O., Saracino D., Proofs and generalizations of a homomesy conjecture of Propp and Roby, Discrete Math. 339 (2016), 194-206, arXiv:1308.0546.
  10. Brouwer A.E., Schrijver A., On the period of an operator, defined on antichains, Mathematisch Centrum, Amsterdam, 1974.
  11. Brundan J., Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006), 17-46, arXiv:math.QA/0509700.
  12. Cameron P.J., Fon-Der-Flaass D.G., Orbits of antichains revisited, European J. Combin. 16 (1995), 545-554.
  13. Chmutov M., Glick M., Pylyavskyy P., The Berenstein-Kirillov group and cactus groups, J. Comb. Algebra 4 (2020), 111-140, arXiv:1609.02046.
  14. Du J., Canonical bases for irreducible representations of quantum ${\rm GL}_n$, Bull. London Math. Soc. 24 (1992), 325-334.
  15. Du J., Canonical bases for irreducible representations of quantum ${\rm GL}_n$. II, J. London Math. Soc. 51 (1995), 461-470.
  16. Einstein D., Propp J., Combinatorial, piecewise-linear, and birational homomesy for products of two chains, arXiv:1310.5294.
  17. Einstein D., Propp J., Piecewise-linear and birational toggling, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), Discrete Math. Theor. Comput. Sci. Proc., AT, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2014, 513-524, arXiv:1404.3455.
  18. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, arXiv:math.AG/0311245.
  19. Fon-Der-Flaass D.G., Orbits of antichains in ranked posets, European J. Combin. 14 (1993), 17-22.
  20. Fontaine B., Kamnitzer J., Cyclic sieving, rotation, and geometric representation theory, Selecta Math. (N.S.) 20 (2014), 609-625, arXiv:1212.1314.
  21. Fraser C., Braid group symmetries of Grassmannian cluster algebras, Selecta Math. (N.S.) 26 (2020), 17, 51 pages, arXiv:1702.00385.
  22. Frieden G., Affine type $A$ geometric crystal on the Grassmannian, J. Combin. Theory Ser. A 167 (2019), 499-563, arXiv:1706.02844.
  23. Fuchs J., Ray U., Schweigert C., Some automorphisms of generalized Kac-Moody algebras, J. Algebra 191 (1997), 518-540, arXiv:q-alg/9605046.
  24. Fuchs J., Schellekens B., Schweigert C., From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), 39-97, arXiv:hep-th/9506135.
  25. Fulton W., Harris J., Representation theory: a first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
  26. Galashin P., Pylyavskyy P., $R$-systems, Selecta Math. (N.S.) 25 (2019), 22, 63 pages, arXiv:1709.00578.
  27. Gansner E.R., On the equality of two plane partition correspondences, Discrete Math. 30 (1980), 121-132.
  28. Garver A., Patrias R., Thomas H., Minuscule reverse plane partitions via quiver representations, Sém. Lothar. Combin. 82B (2020), 44, 12 pages, arXiv:1812.08345.
  29. Gordon B., A proof of the Bender-Knuth conjecture, Pacific J. Math. 108 (1983), 99-113.
  30. Grinberg D., Roby T., Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Combin. 22 (2015), 3.40, 49 pages, arXiv:1402.6178.
  31. Grinberg D., Roby T., Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Combin. 23 (2016), 1.33, 40 pages, arXiv:1402.6178.
  32. Grojnowski I., Lusztig G., A comparison of bases of quantized enveloping algebras, in Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., Providence, RI, 1993, 11-19.
  33. Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497-608, arXiv:1411.1394.
  34. Haiman M.D., Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), 79-113.
  35. Hopkins S., Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion, Experiment. Math., to appear, arXiv:1902.07301.
  36. Jantzen J.C., Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. 67 (1973), v+124 pages.
  37. Karp S.N., Moment curves and cyclic symmetry for positive Grassmannians, Bull. Lond. Math. Soc. 51 (2019), 900-916, arXiv:1805.06004.
  38. Karpman R., Total positivity for the Lagrangian Grassmannian, Adv. in Appl. Math. 98 (2018), 25-76, arXiv:1510.04386.
  39. Karpman R., The purity conjecture in type C, Algebr. Comb. 3 (2020), 1401-1416, arXiv:1907.08275.
  40. Karpman R., Su Y., Combinatorics of symmetric plabic graphs, J. Comb. 9 (2018), 259-278, arXiv:1510.02122.
  41. Kashiwara M., Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455-485.
  42. Kirillov A.N., Berenstein A.D., Groups generated by involutions, Gel'fand-Tsetlin patterns, and combinatorics of Young tableaux, St. Petersburg Math. J. 7 (1996), 77-127.
  43. Knight H., Zelevinsky A., Representations of quivers of type $A$ and the multisegment duality, Adv. Math. 117 (1996), 273-293.
  44. Krattenthaler C., Plane partitions in the work of Richard Stanley and his school, in The Mathematical Legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, 231-261, arXiv:1503.05934.
  45. Kuperberg G., Self-complementary plane partitions by Proctor's minuscule method, European J. Combin. 15 (1994), 545-553, arXiv:math.CO/9411239.
  46. Lam T., Cyclic Demazure modules and positroid varieties, Electron. J. Combin. 26 (2019), 2.28, 20 pages, arXiv:1809.04965.
  47. Lascoux A., Leclerc B., Thibon J.Y., Green polynomials and Hall-Littlewood functions at roots of unity, European J. Combin. 15 (1994), 173-180.
  48. Lenart C., On the combinatorics of crystal graphs. I. Lusztig's involution, Adv. Math. 211 (2007), 204-243, arXiv:math.RT/0509200.
  49. Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498.
  50. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
  51. Macdonald I.G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.
  52. MacMahon P.A., Partitions of numbers whose graphs possess symmetry, Trans. Cambridge Philos. Soc. 17 (1899), 149-170.
  53. MacMahon P.A., Combinatory analysis, Vols. I, II, Dover Phoenix Editions, Dover Publications, Inc., Mineola, NY, 2004.
  54. Musiker G., Roby T., Paths to understanding birational rowmotion on products of two chains, Algebr. Comb. 2 (2019), 275-304, arXiv:1801.03877.
  55. Panyushev D.I., On orbits of antichains of positive roots, European J. Combin. 30 (2009), 586-594, arXiv:0711.3353.
  56. Postnikov A., Total positivity, Grassmannians, and networks, arXiv:math.CO/0609764.
  57. Proctor R.A., Shifted plane partitions of trapezoidal shape, Proc. Amer. Math. Soc. 89 (1983), 553-559.
  58. Proctor R.A., Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin. 5 (1984), 331-350.
  59. Proctor R.A., New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin. 11 (1990), 289-300.
  60. Rao S., Suk J., Dihedral sieving phenomena, Discrete Math. 343 (2020), 111849, 12 pages, arXiv:1710.06517.
  61. Reiner V., Stanton D., White D., The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004), 17-50.
  62. Reiner V., Stanton D., White D., What is ... cyclic sieving?, Notices Amer. Math. Soc. 61 (2014), 169-171.
  63. Rhoades B., Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A 117 (2010), 38-76, arXiv:1005.2568.
  64. Rhoades B., Skandera M., Kazhdan-Lusztig immanants and products of matrix minors, J. Algebra 304 (2006), 793-811.
  65. Rhoades B., Skandera M., Kazhdan-Lusztig immanants and products of matrix minors. II, Linear Multilinear Algebra 58 (2010), 137-150.
  66. Rietsch K., Williams L., Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437-3527, arXiv:1712.00447.
  67. Roby T., Dynamical algebraic combinatorics and the homomesy phenomenon, in Recent trends in combinatorics, IMA Vol. Math. Appl., Vol. 159, Springer, Cham, 2016, 619-652.
  68. Rush D.B., Restriction of global bases and Rhoades's theorem, arXiv:2001.00743.
  69. Rush D.B., Shi X., On orbits of order ideals of minuscule posets, J. Algebraic Combin. 37 (2013), 545-569, arXiv:1108.5245.
  70. Sagan B.E., The cyclic sieving phenomenon: a survey, in Surveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser., Vol. 392, Cambridge University Press, Cambridge, 2011, 183-233, arXiv:1008.0790.
  71. The Sage-Combinat Community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2019, https://combinat.sagemath.org.
  72. The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.4), 2019, https://www.sagemath.org/.
  73. Schützenberger M.P., Quelques remarques sur une construction de Schensted, Math. Scand. 12 (1963), 117-128.
  74. Schützenberger M.P., Promotion des morphismes d'ensembles ordonnés, Discrete Math. 2 (1972), 73-94.
  75. Schützenberger M.P., La correspondance de Robinson, in Combinatoire et représentation du groupe symétrique, Lecture Notes in Math., Vol. 579, Springer-Verlag, Berlin - New York, 1977, 59-113.
  76. Scott J.S., Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), 345-380, arXiv:math.CO/0311148.
  77. Seshadri C.S., Introduction to the theory of standard monomials, 2nd ed., Texts and Readings in Mathematics, Vol. 46, Hindustan Book Agency, New Delhi, 2014.
  78. Sheats J.T., A symplectic jeu de taquin bijection between the tableaux of King and of De Concini, Trans. Amer. Math. Soc. 351 (1999), 3569-3607.
  79. Shen L., Weng D., Cyclic sieving and cluster duality of Grassmannian, SIGMA 16 (2020), 067, 41 pages, arXiv:1803.06901.
  80. Skandera M., On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations, J. Pure Appl. Algebra 212 (2008), 1086-1104.
  81. Stanley R.P., Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), 103-113.
  82. Stanley R.P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  83. Stanley R.P., Promotion and evacuation, Electron. J. Combin. 16 (2009), 9, 24 pages, arXiv:0806.4717.
  84. Stanley R.P., Enumerative combinatorics, 2nd ed., Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge University Press, Cambridge, 2012.
  85. Stembridge J.R., On minuscule representations, plane partitions and involutions in complex Lie groups, Duke Math. J. 73 (1994), 469-490.
  86. Stembridge J.R., Some hidden relations involving the ten symmetry classes of plane partitions, J. Combin. Theory Ser. A 68 (1994), 372-409.
  87. Stembridge J.R., Canonical bases and self-evacuating tableaux, Duke Math. J. 82 (1996), 585-606.
  88. Stier Z., Wellman J., Xu Z., Dihedral sieving on cluster complexes, arXiv:2011.11885.
  89. Striker J., Williams N., Promotion and rowmotion, European J. Combin. 33 (2012), 1919-1942, arXiv:1108.1172.

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