Hyperplane arrangements and diagonal harmonics

In 2003, Haglund’s 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 area’ and bounce statistics of Haglund and Loehr. In this setting, we observe that 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 elementary symmetric functions.

Keywords

Shi arrangement, Ish arrangement, affine permutations, diagonal harmonics, Catalan numbers, nabla operator, parking functions

2010 Mathematics Subject Classification

05E10, 52C35

Full Text (PDF format)

Published 13 August 2013