Hyperplane Arrangements and Diagonal Harmonics

Details Hyperplane Arrangements and Diagonal Harmonics Hyperplane Arrangements and Diagonal Harmonics

2010-05-11

arxiv.org/abs/1005.1949v1

Mathematics

Combinatorics

27 pages, 12 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Hyperplane Arrangements and Diagonal Harmonics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hyperplane Arrangements and Diagonal Harmonics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hyperplane Arrangements and Diagonal Harmonics will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-498495