Ordered set partitions and the 0-Hecke algebra
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 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.

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DOI: 10.5802/alco.10
Classification: 05E10, 05E15
Keywords: Hecke algebra, set partition, coinvariant algebra
Huang, Jia 1; Rhoades, Brendon 2

1 University of Nebraska at Kearney Department of Mathematics Kearney, NE, 68849 (USA)
2 University of California, San Diego Department of Mathematics La Jolla, CA, 92093 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/alco.10/

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