Ordered set partitions and the 0-Hecke algebra
Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 47-80
Let the symmetric group 𝔖 n act on the polynomial ring [x n ]=[x 1 ,,x n ] by variable permutation. The coinvariant algebra is the graded 𝔖 n -module R n :=[x n ]/I n , where I n is the ideal in [x n ] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient R n,k of the polynomial ring [x n ] depending on two positive integers kn which reduces to the classical coinvariant algebra of the symmetric group 𝔖 n when k=n. The quotient R n,k carries the structure of a graded 𝔖 n -module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient S n,k of 𝔽[x n ] which carries a graded action of the 0-Hecke algebra H n (0), where 𝔽 is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case k=n, we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.
Received : 2017-08-09
Revised : 2017-12-07
Accepted : 2017-12-11
Published online : 2018-01-29
DOI : https://doi.org/10.5802/alco.10
Classification:  05E10,  05E15
Keywords: Hecke algebra, set partition, coinvariant algebra
@article{ALCO_2018__1_1_47_0,
     author = {Huang, Jia and Rhoades, Brendon},
     title = {Ordered set partitions and the $0$-Hecke algebra},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {1},
     year = {2018},
     pages = {47-80},
     doi = {10.5802/alco.10},
     zbl = {06882334},
     language = {en},
     url = {http://alco.centre-mersenne.org/item/ALCO_2018__1_1_47_0}
}
Huang, Jia; Rhoades, Brendon. Ordered set partitions and the $0$-Hecke algebra. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 47-80. doi : 10.5802/alco.10. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_47_0/

[1] Adin, R.; Brenti, F.; Roichman, Y. Descent representations and multivariate statistics, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 3051-3082 | Article | MR 2135735 | Zbl 1059.05105

[2] Artin, E. Galois Theory (second edition), Notre Dame Math Lectures, no 2., Notre Dame: University of Notre Dame (1944) | MR 9934 | Zbl 0060.04814

[3] Assem, I.; Simson, D.; Skowroński, A. Elements of the Representation Theory of Associative Algebras. Vol 1., London Mathematical Society Student Texts, Cambridge Univ. Press, Cambridge (2006) | MR 2382332 | Zbl 1092.16001

[4] Berg, C.; Bergeron, N.; Saliola, F.; Serrano, L.; Zabrocki, M. Indecomposable modules for the dual immaculate basis of quasi-symmetric functions, Proc. Amer. Math. Soc., Volume 143 (2015), pp. 991-1000 | Article | MR 3293717 | Zbl 1306.05243

[5] Bergeron, F. Algebraic Combinatorics and Coinvariant Spaces, CMS Treatises in Mathematics. Taylor and Francis, Boca Raton (2009) | MR 2538310 | Zbl 1185.05002

[6] Björner, A.; Wachs, M. L. Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc., Volume 308 (1988), pp. 1-37 | Article | MR 946427 | Zbl 0659.05007

[7] Chan, K.-T. J.; Rhoades, B. Generalized coinvariant algebras for wreath products (Submitted, 2017. arXiv:1701.06256.)

[8] Chevalley, C. Invariants of finite groups generated by reflections, Amer. J. Math., Volume 77 (1955), pp. 778-782 | Article | MR 72877 | Zbl 0065.26103

[9] Cox, D.; Little, J.; O’Shea, D. Ideals, Varieties, and Algorithms (Third edition), Undergraduate Texts in Mathematics. Springer., New York (1992) | MR 1189133 | Zbl 0756.13017

[10] Garsia, A. M. Combinatorial methods in the theory of Cohen–Macaulay rings, Adv. Math., Volume 38 (1980), pp. 229-266 | Article | MR 597728 | Zbl 0461.06002

[11] Garsia, A. M.; Procesi, C. On certain graded S n -modules and the q-Kostka polynomials, Adv. Math., Volume 94 (1992), pp. 82-138 | Article | MR 1168926 | Zbl 0797.20012

[12] Garsia, A. M.; Stanton, D. Group actions on Stanley–Reisner rings and invariants of permutation groups, Adv. Math., Volume 51 (1984), pp. 107-201 | Article | MR 736732 | Zbl 0561.06002

[13] Gessel, I. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra. Contemp. Math., Vol. 34, Amer. Math. Soc., Providence (1984), pp. 289-317 | MR 777705 | Zbl 0562.05007

[14] Grinberg, D.; Reiner, V. Hopf Algebras in Combinatorics (arXiv:1509.8356)

[15] Haglund, J.; Remmel, J.; Wilson, A. T. The Delta conjecture (Accepted, Trans. Amer. Math. Soc., 2016. arXiv:1509.07058) | Zbl 1383.05308

[16] Haglund, J.; Rhoades, B.; Shimozono, M. Ordered set partitions, generalized coinvariant algebras, and the Delta conjecture (Submitted, 2016. arXiv:1509.07058) | Zbl 1384.05043

[17] Huang, J. 0-Hecke actions on coinvariants and flags, J. Algebraic Combin., Volume 40 (2014), pp. 245-278 | Article | MR 3226825 | Zbl 1297.05255

[18] Huang, J. A tableau approach to the representation theory of 0-Hecke algebras, Ann. Comb., Volume 20 (2016), pp. 831-868 | Article | MR 3572389 | Zbl 1354.05140

[19] Krob, D.; Thibon, J.-Y. Noncommutative symmetric functions IV: Quantum linear groups and Hecke actions at q=0, J. Algebraic Combin., Volume 6 (1997), pp. 339-376 | Article | MR 1471894 | Zbl 0881.05120

[20] Macmahon, P. A. Combinatory Analysis (Volume 1), Cambridge University Press, Cambridge (1915) | MR 141605 | Zbl 45.1271.01

[21] Norton, P. N. 0-Hecke algebras, J. Austral. Math. Soc. A, Volume 27 (1979), pp. 337-357 | Article | MR 532754 | Zbl 0407.16019

[22] Remmel, J.; Wilson, A. T. An extension of MacMahon’s Equidistribution Theorem to ordered set partitions, J. Combin. Theory Ser. A, Volume 134 (2015), pp. 242-277 | Article | MR 3345306 | Zbl 1315.05019

[23] Rhoades, B. Ordered set partition statistics and the Delta Conjecture, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 172-217 | Article | MR 3718065 | Zbl 1373.05006

[24] Shephard, G. C.; Todd, J. A. Finite unitary reflection groups, Can. J. Math., Volume 6 (1954), pp. 274-304 | Article | MR 59914 | Zbl 0055.14305

[25] Stanley, R. P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc., Volume 1 (1979), pp. 475-511 | Article | MR 526968 | Zbl 0497.20002

[26] Van Willigenburg, S.; Tewari, V. Modules of the 0-Hecke algebra and quasisymmetric Schur functions, Adv. Math., Volume 285 (2015), pp. 1025-1065 | Article | MR 3406520 | Zbl 1323.05132

[27] Wilson, A. T. An extension of MacMahon’s Equidistribution Theorem to ordered multiset partitions, Electron. J. Combin., Volume 23 (2016), P1.5 | MR 3484710 | Zbl 1329.05030