Séminaire Lotharingien de Combinatoire, 85B.87 (2021), 12 pp.

Brendon Rhoades, Tianyi Yu and Zehong Zhao

Harmonic Bases for Generalized Coinvariant Algebras

Abstract. Let kn be nonnegative integers and let λ be a partition of k. S. Griffin recently introduced a quotient Rn of the polynomial ring Q[x1, ..., xn] in n variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space Vn of harmonics attached to Rn and produce a harmonic basis of Rn indexed by certain ordered set partitions OPn. Our description of Vn involves injective tableaux and Vandermonde determinants and combinatorics of our harmonic basis is governed by a new extension of the Lehmer code of a permutation to OPn.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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