When is a partition, the specialized non-symmetric Macdonald polynomial is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition are multiples of , the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an -fold cyclic shift of the columns. The corresponding CSP polynomial is given by . In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades.
We also introduce a skew version of . We show that these are symmetric and Schur positive via a variant of the Robinson–Schenstedt–Knuth correspondence and we also describe crystal raising and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials.
Revised:
Accepted:
Published online:
Classification: 05E10, 05E05, 06A07
Keywords: Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.
Author's affiliations:
@article{ALCO_2020__3_4_913_0, author = {Alexandersson, Per and Uhlin, Joakim}, title = {Cyclic sieving, skew {Macdonald} polynomials and {Schur} positivity}, journal = {Algebraic Combinatorics}, pages = {913--939}, publisher = {MathOA foundation}, volume = {3}, number = {4}, year = {2020}, doi = {10.5802/alco.123}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.123/} }
TY - JOUR TI - Cyclic sieving, skew Macdonald polynomials and Schur positivity JO - Algebraic Combinatorics PY - 2020 DA - 2020/// SP - 913 EP - 939 VL - 3 IS - 4 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.123/ UR - https://doi.org/10.5802/alco.123 DO - 10.5802/alco.123 LA - en ID - ALCO_2020__3_4_913_0 ER -
Alexandersson, Per; Uhlin, Joakim. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939. doi : 10.5802/alco.123. https://alco.centre-mersenne.org/articles/10.5802/alco.123/
[1] Refined cyclic sieving on words for the major index statistic, European J. Combin., Volume 73 (2018), pp. 37-60 | Article | MR: 3836732 | Zbl: 1393.05263
[2] Non-symmetric Macdonald polynomials and Demazure–Lusztig operators, Sémin. Lothar. Comb., Volume 76 (2019), Paper no. B76d, 27 pages
[3] The cyclic sieving phenomenon on circular Dyck paths, Electron. J. Combin., Volume 26 (2019) no. 4, Paper no. Paper 4.16, 32 pages | Article | MR: 4025420 | Zbl: 1422.05006
[4] LLT polynomials, chromatic quasisymmetric functions and graphs with cycles, Discrete Math., Volume 341 (2018) no. 12, pp. 3453-3482 | Article | MR: 3862644 | Zbl: 1397.05197
[5] Skew characters and cyclic sieving (2020) (https://arxiv.org/abs/2004.01140)
[6] A major-index preserving map on fillings, Electron. J. Combin., Volume 24 (2017) no. 4, Paper no. Paper 4.3, 30 pages | Article | MR: 3711036 | Zbl: 1372.05007
[7] Properties of non-symmetric Macdonald polynomials at and , Ann. Comb., Volume 23 (2019) no. 2, pp. 219-239 | Article | MR: 3962853 | Zbl: 1416.05290
[8] Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials, Trans. Amer. Math. Soc., Volume 370 (2018) no. 12, pp. 8777-8796 | Article | MR: 3864395 | Zbl: 1404.33017
[9] Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials, Sém. Lothar. Combin., Volume 80B (2018), Paper no. Art. 81, 12 pages (30th International Conference on Formal Power Series and Algebraic Combinatorics) | MR: 3940656 | Zbl: 07054676
[10] Combinatorics of Macdonald polynomials and extensions (2007) (https://escholarship.org/uc/item/5zd262sp) (Ph. D. Thesis) | MR: 2710073
[11] Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux, Discrete Math., Volume 319 (2014), pp. 62-67 | Article | MR: 3145263 | Zbl: 1281.05128
[12] Constructions for cyclic sieving phenomena, SIAM J. Discrete Math., Volume 25 (2011) no. 3, pp. 1297-1314 | Article | MR: 2837599 | Zbl: 1237.05209
[13] Haglund’s conjecture on 3-column Macdonald polynomials, Math. Z., Volume 283 (2016) no. 1-2, pp. 601-628 | Article | MR: 3489082 | Zbl: 1344.05147
[14] Crystal bases. Representations and combinatorics, Hackensack, NJ: World Scientific, 2017, xii + 279 pages | Article | Zbl: 06690908
[15] Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., 112, American Mathematical Society, 1994 no. 539, vi+160 pages | Article | MR: 1223236 | Zbl: 0813.05067
[16] Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995) no. 10, pp. 483-515 | Article | MR: 1358032 | Zbl: 0886.05121
[17] Hall–Littlewood functions and Kostka–Foulkes polynomials in representation theory, Sém. Lothar. Combin., Volume 32 (1994), Paper no. Art. B32c, 38 pages | MR: 1399504 | Zbl: 0855.05100
[18] Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997 | Article | Zbl: 0878.14034
[19] -congruences, with applications to supercongruences and the cyclic sieving phenomenon, Int. J. Number Theory, Volume 15 (2019) no. 9, pp. 1919-1968 | Article | MR: 4015520 | Zbl: 1423.11043
[20] The -Catalan numbers and the space of diagonal harmonics, University lecture series, 41, American Mathematical Society, 2007 | MR: 2371044 | Zbl: 1142.05074
[21] A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc., Volume 18 (2005) no. 3, pp. 735-761 | Article | MR: 2138143 | Zbl: 1061.05101
[22] A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math., Volume 130 (2008) no. 2, pp. 359-383 | Article | MR: 2405160 | Zbl: 1246.05162
[23] Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLTpolynomials, Discrete Math., Volume 343 (2020) no. 3, Paper no. 111728, 21 pages | Article | MR: 4033624 | Zbl: 1431.05148
[24] Colorful combinatorics and Macdonald polynomials, European J. Combin., Volume 81 (2019), pp. 354-377 | Article | MR: 3979340 | Zbl: 1420.05180
[25] Various Representations of the Generalized Kostka Polynomials, The Andrews Festschrift, Springer Berlin Heidelberg, Berlin, Heidelberg, 2001, pp. 209-226 | Article
[26] A recursion and a combinatorial formula for Jack polynomials, Invent. Math., Volume 128 (1997) no. 1, pp. 9-22 | Article | MR: 1437493 | Zbl: 0870.05076
[27] Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, Adv. in Appl. Math., Volume 37 (2006) no. 3, pp. 404-431 | Article | MR: 2261181 | Zbl: 1108.05095
[28] Double crystal graphs, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) (Progr. Math.), Volume 210, Birkhäuser Boston, Boston, MA, 2003, pp. 95-114 | Article | MR: 1985724 | Zbl: 1060.05097
[29] Green polynomials and Hall–Littlewood functions at roots of unity, European J. Combin., Volume 15 (1994) no. 2, pp. 173-180 | Article | MR: 1261063 | Zbl: 0789.05093
[30] Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 7, p. A323-A324 | MR: 472993 | Zbl: 0374.20010
[31] Littlewood–Richardson coefficients and Kazhdan–Lusztig polynomials, Combinatorial methods in representation theory (Kyoto, 1998) (Adv. Stud. Pure Math.), Volume 28 (2000), pp. 155-220 | Article | MR: 1864481 | Zbl: 1058.20006
[32] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR: 1354144 | Zbl: 0824.05059
[33] Crystals, semistandard tableaux and cyclic sieving phenomenon, Electron. J. Combin., Volume 26 (2019) no. 4, Paper no. Paper 4.39, 19 pages | Article | MR: 4039345 | Zbl: 1428.05329
[34] Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., Volume 175 (1995) no. 1, pp. 75-121 | Article | MR: 1353018 | Zbl: 0836.43017
[35] The cyclic sieving phenomenon, J. Combin. Theory Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | Article | MR: 2087303 | Zbl: 1052.05068
[36] Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A, Volume 117 (2010) no. 1, pp. 38-76 | Article | MR: 2557880 | Zbl: 1230.05289
[37] Hall–Littlewood polynomials and fixed point enumeration, Discrete Math., Volume 310 (2010) no. 4, pp. 869-876 | Article | MR: 2574838 | Zbl: 1228.05294
[38] Cyclic sieving and plethysm coefficients, Trans. Amer. Math. Soc., Volume 371 (2019) no. 2, pp. 923-947 | Article | MR: 3885166 | Zbl: 1401.05309
[39] Congruence properties of -analogs, Adv. Math., Volume 95 (1992) no. 1, pp. 127-143 | Article | MR: 1176155 | Zbl: 0761.11010
[40] The cyclic sieving phenomenon: a survey, Surveys in combinatorics 2011 (London Math. Soc. Lecture Note Ser.), Volume 392, Cambridge Univ. Press, Cambridge, 2011, pp. 183-233 | Article | MR: 2866734 | Zbl: 1233.05028
[41] Cyclic Sieving and Cluster Duality for Grassmannian (2018) (https://arxiv.org/abs/1803.06901)
[42] Crystals for dummies, Online, 2005 (https://www.aimath.org/WWN/kostka/crysdumb.pdf)
[43] Graded characters of modules supported in the closure of a nilpotent conjugacy class, European J. Combin., Volume 21 (2000) no. 2, pp. 257-288 | Article | MR: 1742440 | Zbl: 0956.05100
[44] The On-Line Encyclopedia of Integer Sequences, Online, 2019 (https://oeis.org) | Zbl: 06989892
[45] Enumerative Combinatorics: Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 2001 | Zbl: 0978.05002
[46] A local characterization of simply-laced crystals, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, pp. 4807-4823 | Article | MR: 1997585 | Zbl: 1047.17007
[47] A -Analog of Schur’s Q-Functions, Algebraic Combinatorics and Quantum Groups (Jing, Naihuan, ed.), World Scientific, 2003, pp. 135-161 | Article | Zbl: 1076.05523
[48] Combinatorics of Macdonald polynomials and cyclic sieving (2019) (http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1282825) (M.S Thesis)
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