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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 *R*_{n,λ} of the polynomial ring
**Q**[*x*_{1}, ..., *x*_{n}] 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 *V*_{n,λ} of harmonics attached to *R*_{n,λ}
and produce a harmonic basis of *R*_{n,λ} indexed by certain ordered set partitions *OP*_{n,λ}.
Our description of *V*_{n,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 *OP*_{n,λ}.

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

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