Hyperplane Arrangements and Diagonal Harmonics

TL;DR

This paper introduces a new combinatorial interpretation of the $q,t$-Catalan numbers and diagonal harmonics using affine Weyl groups, defining new statistics related to hyperplane arrangements and extending these concepts to conjecture interpretations for powers of the nabla operator.

Contribution

It proposes a novel combinatorial framework involving affine permutations and hyperplane arrangements, connecting to Haglund's bounce statistic and extending to conjectural interpretations of nabla operator powers.

Findings

Defined two new statistics on affine permutations related to hyperplane arrangements.

Proved the equivalence of these statistics to Haglund and Loehr's area' and bounce statistics.

Extended the framework to conjecturally interpret all powers of the nabla operator.

Abstract

In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\sf area'} and {\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that {\sf bounce} is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended" Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to the elementary symmetric functions.