[Paper Review] Hyperplane Arrangements and Diagonal Harmonics

Drew Armstrong|arXiv (Cornell University)|May 11, 2010

Advanced Combinatorial Mathematics1 references4 citations

TL;DR

This paper proposes a new combinatorial interpretation of the $q,t$-Catalan numbers and the bigraded Hilbert series of diagonal harmonics using the affine Weyl group of type $A$. It introduces two statistics—'area' and 'bounce'—on affine permutations via the Shi and Ish hyperplane arrangements, proving they match Haglund and Loehr's original statistics. The key contribution is a geometric and algebraic framework linking hyperplane arrangements to diagonal harmonics through the root lattice and affine permutations.

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.

Motivation & Objective

• To provide a new combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics using affine Weyl group structures.
• To define and analyze two statistics—'area' and 'bounce'—on affine permutations arising from the Shi and Ish hyperplane arrangements.
• To establish a connection between the bounce statistic and the root lattice, offering a geometric interpretation of the $q,t$-Catalan generating function.
• To extend these statistics to extended Shi arrangements and bounded chambers, suggesting a conjectural interpretation for higher powers of the Bergeron-Garsia nabla operator.
• To lay the groundwork for generalizing these results beyond type $A$, particularly in other Weyl groups and complex types where no such combinatorial interpretation currently exists.

Proposed method

• Define the Ish arrangement as a new hyperplane arrangement in $×_{0}^{n}$, dual to the Shi arrangement, using hyperplanes $e_i - e_j = 1$ for $i < j$, and show it partitions the space into alcoves.
• Introduce the 'ish' statistic on affine permutations as the inverse of the number of hyperplanes in the Ish arrangement crossed when moving from the fundamental alcove to a given alcove.
• Define the 'shi' statistic on affine permutations as the number of hyperplanes in the Shi arrangement crossed in the same path, and prove it matches Haglund's bounce statistic.
• Establish that the 'area' statistic (defined as the sum of coordinates of the affine permutation) corresponds to the standard area statistic in the $q,t$-Catalan expansion.
• Use the action of the affine Weyl group $χτ(n)$ to act simply transitively on alcoves, enabling a bijection between affine permutations and chambers in the arrangement.
• Prove that the generating function $F(p,n;q,t) = \sum_{A \subseteq D^p(n)} q^{\text{ish}^{-1}(A)} t^{(p-1)(n-1)/2 - \text{stat}(A)}$ satisfies symmetry and specialization properties, including $F(p,n;q,1/q) = [p]_q^{n-1}$ and $F_+(p,n;q,t) = \frac{1}{[p+n]_q} \binom{p+n}{n}_q$.

Experimental results

Research questions

• RQ1 Can the $q,t$-Catalan numbers be interpreted combinatorially via the affine Weyl group of type $A$ using hyperplane arrangements?
• RQ2 Is the bounce statistic naturally expressible as a statistic on the root lattice through the Ish arrangement?
• RQ3 Can the statistics 'area' and 'bounce' on affine permutations be defined such that they recover Haglund and Loehr's original statistics?
• RQ4 What is the generalization of the 'shi' statistic to higher-dimensional simplices $D^p(n)$, particularly for $p=2$, $n=5$?
• RQ5 Can the framework be extended to other types beyond $A$, especially in the context of rational Cherednik algebras and the nabla operator?

Key findings

• The 'ish' statistic on affine permutations, defined via the Ish arrangement, matches the area statistic in Haglund's $q,t$-Catalan formula.
• The 'shi' statistic on affine permutations, defined via the Shi arrangement, is equivalent to the bounce statistic in the original combinatorial interpretation of the $q,t$-Catalan numbers.
• The bounce statistic is naturally expressed as a function on the root lattice, providing a geometric interpretation of the $t$-degree in the diagonal harmonics Hilbert series.
• For the simplex $D^2(5)$, the generating function of the 'ish' statistic over all 16 alcoves is $10 + 5q + q^2$, and over the 3 alcoves with inverse in the dominant cone, it is $1 + q + q^2$, supporting the conjectured symmetry.
• The generating function $F(p,n;q,t)$ satisfies $F(p,n;q,t) = F(p,n;t,q)$ and $q^{(p-1)(n-1)/2}F(p,n;q,1/q) = [p]_q^{n-1}$, confirming key properties of the $q,t$-Catalan numbers.
• The paper conjectures that the generating function $F(p,n;q,t)$ generalizes the $q,t$-Catalan to all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions.