Harmonic bases for generalized coinvariant algebras

TL;DR

This paper introduces a harmonic basis for a new class of quotient rings generalizing coinvariant rings and Springer fiber cohomology, using extended Lehmer code combinatorics.

Contribution

It constructs a harmonic basis for the quotient rings $R_{n,}$, extending Lehmer code combinatorics to ordered set partitions, and links algebraic and combinatorial structures.

Findings

Harmonic basis indexed by ordered set partitions

Extension of Lehmer code to new combinatorial objects

Connection to generalized coinvariant algebras

Abstract

Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ 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 $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. The combinatorics of this basis is governed by a new extension of the {\em Lehmer code} of a permutation to $\mathcal{OP}_{n, \lambda}$.