Harmonic bases for generalized coinvariant algebras

Abstract

Let $k\le n$ be nonnegative integers and let $\hat{I}»$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\hat{I}»}$ 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 $V_{n,\hat{I}»}$ of harmonics attached to $R_{n,\hat{I}»}$ and produce a harmonic basis of $R_{n,\hat{I}»}$ indexed by certain ordered set partitions ${\mathcal{O}\mathcal{P}}_{n,\hat{I}»}$. The combinatorics of this basis is governed by a new extension of the {\em Lehmer code} of a permutation to ${\mathcal{O}\mathcal{P}}_{n,\hat{I}»}$.

18 pages