Harmonic bases for generalized coinvariant algebras

Brendon Rhoades ; Tianyi Yu ; Zehong Zhao

The electronic journal of combinatorics, Tome 27 (2020) no. 4

Résumé

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 Lehmer code of a permutation to $\mathcal{OP}_{n, \lambda}$.

Détail

DOI : 10.37236/9610

Classification : 05E05, 05E10, 05E40, 20C30, 05A18
Mots-clés : delta conjecture, Lehmer code