Harmonic Bases for Generalized Coinvariant Algebras

  • Brendon Rhoades
  • Tianyi Yu
  • Zehong Zhao


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 $\mathcal{OP}_{n, \lambda}$.

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