TY - JOUR
T1 - Ordered set partitions and the 0-Hecke algebra
AU - Huang, Jia
AU - Rhoades, Brendon
N1 - Funding Information:
Acknowledgements. B. Rhoades was partially supported by NSF Grant DMS-1500838.
Publisher Copyright:
© Algebraic Combinatorics 2018.
PY - 2018
Y1 - 2018
N2 - Let the symmetric group Snact on the polynomial ring Q[xn] = Q[x1, . . . , xn] by variable permutation. The coinvariant algebra is the graded Sn-module Rn:= Q[xn]/In, where In is the ideal in Q[xn] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient Rn,kof the polynomial ring Q[xn] depending on two positive integers k ≤ n which reduces to the classical coinvariant algebra of the symmetric group Snwhen k = n. The quotient Rn,kcarries the structure of a graded Sn-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 Sn,kof F[xn] which carries a graded action of the 0-Hecke algebra Hn(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.
AB - Let the symmetric group Snact on the polynomial ring Q[xn] = Q[x1, . . . , xn] by variable permutation. The coinvariant algebra is the graded Sn-module Rn:= Q[xn]/In, where In is the ideal in Q[xn] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient Rn,kof the polynomial ring Q[xn] depending on two positive integers k ≤ n which reduces to the classical coinvariant algebra of the symmetric group Snwhen k = n. The quotient Rn,kcarries the structure of a graded Sn-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 Sn,kof F[xn] which carries a graded action of the 0-Hecke algebra Hn(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.
KW - Coinvariant algebra
KW - Hecke algebra
KW - Set partition
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U2 - 10.5802/alco.10
DO - 10.5802/alco.10
M3 - Article
AN - SCOPUS:85117773799
SN - 2589-5486
VL - 1
SP - 47
EP - 80
JO - Algebraic Combinatorics
JF - Algebraic Combinatorics
IS - 1
ER -