Séminaire Lotharingien de Combinatoire, 85B.87 (2021), 12 pp.
Brendon Rhoades, Tianyi Yu and
Zehong Zhao
Harmonic Bases for Generalized Coinvariant Algebras
Abstract.
Let k ≤ n be nonnegative integers and let λ be a partition of k.
S. Griffin recently introduced a quotient Rn,λ 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,λ of harmonics attached to Rn,λ
and produce a harmonic basis of Rn,λ indexed by certain ordered set partitions OPn,λ.
Our description of Vn,λ 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,λ.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
The following versions are available: