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
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},
     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, p. 47-80. doi : 10.5802/alco.10. http://alco.centre-mersenne.org/item/ALCO_2018__1_1_47_0/

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