Let the symmetric group act on the polynomial ring by variable permutation. The coinvariant algebra is the graded -module , where is the ideal in generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient of the polynomial ring depending on two positive integers which reduces to the classical coinvariant algebra of the symmetric group when . The quotient carries the structure of a graded -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 of which carries a graded action of the 0-Hecke algebra , where is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case , we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.10
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}, pages = {47--80}, publisher = {MathOA foundation}, volume = {1}, number = {1}, year = {2018}, doi = {10.5802/alco.10}, zbl = {06882334}, mrnumber = {3857159}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.10/} }
TY - JOUR AU - Huang, Jia AU - Rhoades, Brendon TI - Ordered set partitions and the $0$-Hecke algebra JO - Algebraic Combinatorics PY - 2018 SP - 47 EP - 80 VL - 1 IS - 1 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.10/ DO - 10.5802/alco.10 LA - en ID - ALCO_2018__1_1_47_0 ER -
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. http://archive.numdam.org/articles/10.5802/alco.10/
[1] Descent representations and multivariate statistics, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 3051-3082 | DOI | MR | Zbl
[2] Galois Theory (second edition), Notre Dame Math Lectures, no 2., Notre Dame: University of Notre Dame, 1944 | MR | Zbl
[3] Elements of the Representation Theory of Associative Algebras. Vol 1., London Mathematical Society Student Texts, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl
[4] Indecomposable modules for the dual immaculate basis of quasi-symmetric functions, Proc. Amer. Math. Soc., Volume 143 (2015), pp. 991-1000 | DOI | MR | Zbl
[5] Algebraic Combinatorics and Coinvariant Spaces, CMS Treatises in Mathematics. Taylor and Francis, Boca Raton, 2009 | DOI | MR | Zbl
[6] Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc., Volume 308 (1988), pp. 1-37 | DOI | MR | Zbl
[7] Generalized coinvariant algebras for wreath products (Submitted, 2017. arXiv:1701.06256.)
[8] Invariants of finite groups generated by reflections, Amer. J. Math., Volume 77 (1955), pp. 778-782 | DOI | MR | Zbl
[9] Ideals, Varieties, and Algorithms (Third edition), Undergraduate Texts in Mathematics. Springer., New York, 1992 | MR | Zbl
[10] Combinatorial methods in the theory of Cohen–Macaulay rings, Adv. Math., Volume 38 (1980), pp. 229-266 | DOI | MR | Zbl
[11] On certain graded -modules and the -Kostka polynomials, Adv. Math., Volume 94 (1992), pp. 82-138 | DOI | MR | Zbl
[12] Group actions on Stanley–Reisner rings and invariants of permutation groups, Adv. Math., Volume 51 (1984), pp. 107-201 | DOI | MR | Zbl
[13] 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 | DOI | MR | Zbl
[14] Hopf Algebras in Combinatorics (arXiv:1509.8356)
[15] The Delta conjecture (Accepted, Trans. Amer. Math. Soc., 2016. arXiv:1509.07058) | DOI | Zbl
[16] Ordered set partitions, generalized coinvariant algebras, and the Delta conjecture (Submitted, 2016. arXiv:1509.07058) | DOI | Zbl
[17] -Hecke actions on coinvariants and flags, J. Algebraic Combin., Volume 40 (2014), pp. 245-278 | DOI | MR | Zbl
[18] A tableau approach to the representation theory of -Hecke algebras, Ann. Comb., Volume 20 (2016), pp. 831-868 | DOI | MR | Zbl
[19] Noncommutative symmetric functions IV: Quantum linear groups and Hecke actions at , J. Algebraic Combin., Volume 6 (1997), pp. 339-376 | DOI | MR | Zbl
[20] Combinatory Analysis (Volume 1), Cambridge University Press, Cambridge, 1915 | MR | Zbl
[21] -Hecke algebras, J. Austral. Math. Soc. A, Volume 27 (1979), pp. 337-357 | DOI | MR | Zbl
[22] An extension of MacMahon’s Equidistribution Theorem to ordered set partitions, J. Combin. Theory Ser. A, Volume 134 (2015), pp. 242-277 | DOI | MR | Zbl
[23] Ordered set partition statistics and the Delta Conjecture, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 172-217 | DOI | MR | Zbl
[24] Finite unitary reflection groups, Can. J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl
[25] Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc., Volume 1 (1979), pp. 475-511 | DOI | MR | Zbl
[26] Modules of the -Hecke algebra and quasisymmetric Schur functions, Adv. Math., Volume 285 (2015), pp. 1025-1065 | DOI | MR | Zbl
[27] An extension of MacMahon’s Equidistribution Theorem to ordered multiset partitions, Electron. J. Combin., Volume 23 (2016), P1.5 | MR | Zbl
Cited by Sources: