Cyclic sieving, skew Macdonald polynomials and Schur positivity
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939.

When λ is a partition, the specialized non-symmetric Macdonald polynomial E λ (x;q;0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by E λ (x;q;0). 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 E λ (x;q;0). 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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.123
Classification: 05E10,  05E05,  06A07
Keywords: Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.
@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
AU  - Alexandersson, Per
AU  - Uhlin, Joakim
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] Ahlbach, Connor; Swanson, Joshua P. 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] Alexandersson, Per Non-symmetric Macdonald polynomials and Demazure–Lusztig operators, Sémin. Lothar. Comb., Volume 76 (2019), Paper no. B76d, 27 pages

[3] Alexandersson, Per; Linusson, Svante; Potka, Samu 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] Alexandersson, Per; Panova, Greta 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] Alexandersson, Per; Pfannerer, Stephan; Rubey, Martin; Uhlin, Joakim Skew characters and cyclic sieving (2020) (https://arxiv.org/abs/2004.01140)

[6] Alexandersson, Per; Sawhney, Mehtaab 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] Alexandersson, Per; Sawhney, Mehtaab Properties of non-symmetric Macdonald polynomials at q=1 and q=0, Ann. Comb., Volume 23 (2019) no. 2, pp. 219-239 | Article | MR 3962853 | Zbl 1416.05290

[8] Assaf, Sami 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] Assaf, Sami; González, Nicolle S. 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] Bandlow, Jason Combinatorics of Macdonald polynomials and extensions (2007) (https://escholarship.org/uc/item/5zd262sp) (Ph. D. Thesis) | MR 2710073

[11] Bennett, Max; Madill, Blake; Stokke, Anna 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] Berget, Andrew; Eu, Sen-Peng; Reiner, Victor Constructions for cyclic sieving phenomena, SIAM J. Discrete Math., Volume 25 (2011) no. 3, pp. 1297-1314 | Article | MR 2837599 | Zbl 1237.05209

[13] Blasiak, Jonah 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] Bump, Daniel; Schilling, Anne Crystal bases. Representations and combinatorics, Hackensack, NJ: World Scientific, 2017, xii + 279 pages | Article | Zbl 06690908

[15] Butler, Lynne M. 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] Cherednik, Ivan Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995) no. 10, pp. 483-515 | Article | MR 1358032 | Zbl 0886.05121

[17] Désarménien, Jacques; Leclerc, Bernard; Thibon, Jean-Yves 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] Fulton, William Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997 | Article | Zbl 0878.14034

[19] Gorodetsky, Ofir q-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] Haglund, James The q,t-Catalan numbers and the space of diagonal harmonics, University lecture series, 41, American Mathematical Society, 2007 | MR 2371044 | Zbl 1142.05074

[21] Haglund, James; Haiman, Mark; Loehr, Nicholas A. 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] Haglund, James; Haiman, Mark; Loehr, Nicholas A. 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] Huh, JiSun; Nam, Sun-Young; Yoo, Meesue 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] Kaliszewski, Ryan; Morse, Jennifer Colorful combinatorics and Macdonald polynomials, European J. Combin., Volume 81 (2019), pp. 354-377 | Article | MR 3979340 | Zbl 1420.05180

[25] Kirillov, Anatol N.; Schilling, Anne; Shimozono, Mark Various Representations of the Generalized Kostka Polynomials, The Andrews Festschrift, Springer Berlin Heidelberg, Berlin, Heidelberg, 2001, pp. 209-226 | Article

[26] Knop, Friedrich; Sahi, Siddhartha 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] Krattenthaler, Christian 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] Lascoux, Alain 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] Lascoux, Alain; Leclerc, Bernard; Thibon, Jean-Yves 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] Lascoux, Alain; Schützenberger, Marcel-Paul 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] Leclerc, Bernard; Thibon, Jean-Yves 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] Macdonald, Ian G. 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] Oh, Young-Tak; Park, Euiyong 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] Opdam, Eric M. 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] Reiner, Victor; Stanton, Dennis; White, Dennis E. The cyclic sieving phenomenon, J. Combin. Theory Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | Article | MR 2087303 | Zbl 1052.05068

[36] Rhoades, Brendon 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] Rhoades, Brendon Hall–Littlewood polynomials and fixed point enumeration, Discrete Math., Volume 310 (2010) no. 4, pp. 869-876 | Article | MR 2574838 | Zbl 1228.05294

[38] Rush, David B. Cyclic sieving and plethysm coefficients, Trans. Amer. Math. Soc., Volume 371 (2019) no. 2, pp. 923-947 | Article | MR 3885166 | Zbl 1401.05309

[39] Sagan, Bruce E. Congruence properties of q-analogs, Adv. Math., Volume 95 (1992) no. 1, pp. 127-143 | Article | MR 1176155 | Zbl 0761.11010

[40] Sagan, Bruce E. 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] Shen, Linhui; Weng, Daping Cyclic Sieving and Cluster Duality for Grassmannian (2018) (https://arxiv.org/abs/1803.06901)

[42] Shimozono, Mark Crystals for dummies, Online, 2005 (https://www.aimath.org/WWN/kostka/crysdumb.pdf)

[43] Shimozono, Mark; Weyman, Jerzy 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] Sloane, Neil J. A. The On-Line Encyclopedia of Integer Sequences, Online, 2019 (https://oeis.org) | Zbl 06989892

[45] Stanley, Richard P. Enumerative Combinatorics: Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 2001 | Zbl 0978.05002

[46] Stembridge, John R. 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] Tudose, Geanina; Zabrocki, Mike A q-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] Uhlin, Joakim Combinatorics of Macdonald polynomials and cyclic sieving (2019) (http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1282825) (M.S Thesis)

Cited by Sources: