Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson, Per; Uhlin, Joakim

doi:10.5802/alco.123

Alexandersson, Per ¹; Uhlin, Joakim ¹

¹ Dept. of Mathematics Royal Institute of Technology SE-100 44 Stockholm, Sweden

Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939.

Abstract

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: 2019-08-15
Revised: 2020-03-25
Accepted: 2020-04-12
Published online: 2020-08-20

DOI: 10.5802/alco.123

Classification: 05E10, 05E05, 06A07
Keywords: Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.

Author's affiliations:

Alexandersson, Per ¹; Uhlin, Joakim ¹

¹ Dept. of Mathematics Royal Institute of Technology SE-100 44 Stockholm, Sweden

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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
SP  - 913
EP  - 939
VL  - 3
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.123/
DO  - 10.5802/alco.123
LA  - en
ID  - ALCO_2020__3_4_913_0
ER  -

%0 Journal Article
%A Alexandersson, Per
%A Uhlin, Joakim
%T Cyclic sieving, skew Macdonald polynomials and Schur positivity
%J Algebraic Combinatorics
%D 2020
%P 913-939
%V 3
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.123/
%R 10.5802/alco.123
%G en
%F ALCO_2020__3_4_913_0

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/

References
Cited by

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[4] Alexandersson, Per; Panova, Greta LLT polynomials, chromatic quasisymmetric functions and graphs with cycles, Discrete Math., Volume 341 (2018) no. 12, pp. 3453-3482 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[10] Bandlow, Jason Combinatorics of Macdonald polynomials and extensions, Ph. D. Thesis, UC San Diego (2007) (https://escholarship.org/uc/item/5zd262sp) | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[13] Blasiak, Jonah Haglund’s conjecture on 3-column Macdonald polynomials, Math. Z., Volume 283 (2016) no. 1-2, pp. 601-628 | DOI | MR | Zbl

[14] Bump, Daniel; Schilling, Anne Crystal bases. Representations and combinatorics, Hackensack, NJ: World Scientific, 2017, xii + 279 pages | DOI | Zbl

[15] Butler, Lynne M. Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., 112, American Mathematical Society, 1994 no. 539, vi+160 pages | DOI | MR | Zbl

[16] Cherednik, Ivan Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995) no. 10, pp. 483-515 | DOI | MR | Zbl

[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 | Zbl

[18] Fulton, William Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997 | DOI | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[24] Kaliszewski, Ryan; Morse, Jennifer Colorful combinatorics and Macdonald polynomials, European J. Combin., Volume 81 (2019), pp. 354-377 | DOI | MR | Zbl

[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 | DOI

[26] Knop, Friedrich; Sahi, Siddhartha A recursion and a combinatorial formula for Jack polynomials, Invent. Math., Volume 128 (1997) no. 1, pp. 9-22 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[34] Opdam, Eric M. Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., Volume 175 (1995) no. 1, pp. 75-121 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[36] Rhoades, Brendon Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A, Volume 117 (2010) no. 1, pp. 38-76 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[44] Sloane, Neil J. A. The On-Line Encyclopedia of Integer Sequences, Online, 2019 (https://oeis.org) | Zbl

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

[46] Stembridge, John R. A local characterization of simply-laced crystals, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, pp. 4807-4823 | DOI | MR | Zbl

[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 | DOI | Zbl

[48] Uhlin, Joakim Combinatorics of Macdonald polynomials and cyclic sieving, M.S Thesis, KTH, Mathematics (Div.), January (2019) (http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1282825)

Cited by Sources: