Hyperplane arrangements and diagonal harmonics

Research output: Contribution to conferencePaperpeer-review

4 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages39-50
Number of pages12
StatePublished - Dec 1 2011
Event23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 - Reykjavik, Iceland
Duration: Jun 13 2011Jun 17 2011

Other

Other23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11
CountryIceland
CityReykjavik
Period6/13/116/17/11

Keywords

  • Affine permutations
  • Catalan numbers
  • Diagonal harmonics
  • Ish arrangement
  • Nabla operator
  • Parking functions
  • Shi arrangement

ASJC Scopus subject areas

  • Algebra and Number Theory

Fingerprint Dive into the research topics of 'Hyperplane arrangements and diagonal harmonics'. Together they form a unique fingerprint.

Cite this