Diagonal Harmonics Research Articles

Triangular Diagonal Harmonics Conjectures

The purpose of this paper is to present conjectures that extend to any “triangular” partitions (partitions “under any line” in the terminology of Blasiak-Haiman-Morse-Pun-Seelinger), properties of the Frobenius transform of multivariate diagonal harmonics modules. In their simplest version, these last modules correspond to the special case of “staircase” partitions, that is of the form ( n − 1 , n − 2 , … , 1 ) . The conjectures are motivated and supported both by extensive experimental computer algebra calculations, as well as stability properties.

Chromatic symmetric functions of Dyck paths and [formula omitted]-rook theory

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian–Wachs q-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley and Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu and Nigro (2021) and Guay-Paquet (private communication), using q-hit numbers. Among our main results is a new proof of Guay-Paquet’s elegant identity expressing the q-CSFs in a CSF basis with q-hit coefficients. We further show its equivalence to the Abreu–Nigro identity expanding the q-CSF in the elementary symmetric functions. In the course of our work we establish that the q-hit numbers in these expansions differ from the originally assumed Garsia–Remmel q-hit numbers by certain powers of q. We prove new identities for these q-hit numbers, and establish connections between the three different variants.

Hopf dreams and diagonal harmonics

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday–Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on ν $\nu$ -trees related to ν $\nu$ -Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics.

The valley version of the Extended Delta Conjecture

The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics, and the Delta Conjecture of Haglund, Remmel and the second author provides two generalizations of the Shuffle Theorem to the delta operator expression Δek′en. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression Δek′Δhren, which is analogous to the rise version of the Delta Conjecture. Recently, D'Adderio, Iraci and Wyngaerd proved the rise version of the Extended Delta Conjecture at the case when t=0. In this paper, we propose a new valley version of the Extended Delta Conjecture. Then, we work on the combinatorics of extended ordered multiset partitions to prove that the two conjectures for Δek′Δhren are equivalent when t or q equals 0, thus proving the valley version of the Extended Delta Conjecture when t or q equals 0.

The Steep-Bounce zeta map in Parabolic Cataland

As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorial object called “left-aligned colorable tree”, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in q,t-Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.

A Proof of the Delta Conjecture When $$\varvec{q=0}$$

In Haglund et al. (Trans. Amer. Math. Soc. 370(6):4029–4057, 2018), Haglund, Remmel and Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials (Garsia and Haiman in J. Algebraic Combin. 5:191–244, 1996; Haiman in Invent. Math. 149:371–407, 2002). The Delta Conjecture involves two parameters q, t; in this article we give the first proof that the Delta Conjecture is true when $$q=0$$ or $$t=0$$ .

Flow Polytopes and the Space of Diagonal Harmonics

AbstractA result of Haglund implies that the$(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a$(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector$(-n,1,\ldots ,1)$. We study the$(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at$t=1$,$0$, and$q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the$(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

On the Poset and Asymptotics of Tesler Matrices

Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $q-1$. We use a method of Hallam and Sagan to prove a stronger version of this conjecture for posets of a certain class of generalized Tesler matrices. We also study bounds for the number of Tesler matrices and how they compare to the number of parking functions, the dimension of the space of diagonal harmonics.

On Parking Functions and the Zeta Map in Types B, C and D

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

Ordered set partition statistics and the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund–Remmel–Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.

Generalized Polarization Modules

This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\leq i\leq \ell$ et $1\leq j\leq n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n\times{GL}_{\ell}(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in $n$ variables, for every $n\geq 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$.

A Polynomial Expression for the Character of Diagonal Harmonics

Based on his study of the Hilbert scheme from algebraic geometry, Haiman [Invent. Math. 149 (2002), pp. 371–407] obtained a formula for the character of the space of diagonal harmonics under the diagonal action of the symmetric group, as a sum of Macdonald polynomials with rational coefficients. In this paper we show how Haiman's formula, combined with identities involving plethystic symmetric function operators, yields a new formula for this character. Our formula doesn't involve any mention of Macdonald polynomials, and the coefficients are visibly polynomials (not rational functions), although they are not manifestly positive. Our formula can be expressed as either a sum of weighted Tesler matrices, or as the constant term in a multivariate Laurent series.

Generalized Polarization Modules (extended abstract)

This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$. Ce travail s'inscrit dans la lignée de recherche des travaux de M. Haiman sur le théorème de l'opérateur (ex-conjecture de l'opérateur). Ce théorème affirme que le plus petit $\mathfrak{S}_n$-module clos par dérivation partielle et clos par l'action des opérateurs de polarisation qui contient le déterminant de Vandermonde est l'espace des polynômes harmoniques diagonaux. On commence par généraliser le contexte du théorème de l'opérateur au contexte de polynômes à ensembles de $n$ variables $x_{ij}$ avec $1\le i \le \ell$ et $1 \le j \le n$. Étant donnée une famille $\mathfrak{S}_n$-stable $F$ des polynômes homogènes en les variables $x_{ij}$, le plus petit espace vectoriel $\mathcal{M}_F$ clos par dérivation partielle et clos par léaction des opérateurs de polarisation contenant $F$ est le module de polarisation engendré par la famille $F$. Les modules $\mathcal{M}_F$ sont tous des représentations du produit direct $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. Dans le but d'étudier la décomposition en sous-modules irréductibles on calcule la caractéristique de Frobenius graduée de ces modules. Pour plusieurs cas de familles homogènes $\mathfrak{S}_n$-stables constituées des polynômes homogènes à $n$ variables, pour tout $n \ge 1$, on démontre des formules générales pour cette caractéristique graduée de façon globale, indépendante de la valeur de $\ell$.

Generalized Tesler matrices, virtual Hilbert series, and Macdonald polynomial operators

We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses <i>virtual Hilbert series</i>, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively. Nous généralisons les définitions précédentes de matrices Tesler pour permettre les entrées de la matrice négatives et des montants crochet non-positifs. Notre principal résultat est une interprétation algébrique d’une certaine somme pondérée sur ces matrices. Notre interprétation utilise <i>série de Hilbert virtuel</i>, une nouvelle classe de spécialisations fonctionnelles symétriques qui sont définies par leurs valeurs sur les polynômes de Macdonald (modifiées). À la suite de cette interprétation, on obtient une expression de la matrice Tesler pour la salle intérieure produit $\langle \Delta_f e_n, p_{1^{n}}\rangle$, où $\Delta_f$ est un opérateur de fonction symétrique de la théorie harmonique de diagonale. Nous utilisons notre expression de la matrice Tesler, ainsi que divers faits sur des matrices Tesler, de fournir des formules simples pour $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ et $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ impliquant $q; t$-coefficients binomial et ensemble ordonné partitions, respectivement.

Parking spaces

Let W be a Weyl group with root lattice Q and Coxeter number h. The elements of the finite torus Q/(h+1)Q are called the W-parking functions, and we call the permutation representation of W on the set of W-parking functions the (standard) W-parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W-parking spaces, called the noncrossing parking space and the algebraic parking space, with the following features:•They are defined more generally for real reflection groups.•They carry not just W-actions, but W×C-actions, where C is the cyclic subgroup of W generated by a Coxeter element.•In the crystallographic case, both are isomorphic to the standard W-parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of W×C. This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. Even the weakest form of the Main Conjecture has interesting combinatorial consequences, and this weak form is proven in all types except E7 and E8. We provide evidence for the stronger forms of the conjecture, including proofs in some cases, and suggest further directions for the theory.

New combinatorial formulations of the shuffle conjecture

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function ∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We also give new versions of some recently proposed combinatorial formulas for ∇(Cα) and ∇(s(k,1(n−k))), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund's bounce statistic.

Statistics on parallelogram polyominoes and a [formula omitted]-analogue of the Narayana numbers

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area,bounce) and (area,dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

Bijections for the Shi and Ish arrangements

The Shi hyperplane arrangementShi(n) was introduced by Shi to study the Kazhdan–Lusztig cellular structure of the affine symmetric group. The Ish hyperplane arrangementIsh(n) was introduced by Armstrong in the study of diagonal harmonics. Armstrong and Rhoades discovered a deep combinatorial similarity between the Shi and Ish arrangements. We solve a collection of problems posed by Armstrong (2013) and Armstrong and Rhoades (2012) by giving bijections between regions of Shi(n) and Ish(n) which preserve certain statistics. Our bijections generalize to the ‘deleted arrangements’ Shi(G) and Ish(G) which depend on a graph G on n vertices. The key tools in our bijections are the introduction of an Ish analog of parking functions called rook words and a new instance of the cycle lemma of enumerative combinatorics.

Hyperplane arrangements and diagonal harmonics

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.

A $q,t-$analogue of Narayana numbers

We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.