2024-01-19

For background, see the canonical sources by J. Haglund [Hag07], and F. Bergeron[Ber09]. However, there has been a lot of exciting new development since the writing of these books.

## Introduction

The ring of diagonal harmonics is defined as the set of polynomials in $\setC[X,Y],$ such that for all $a+b \ge 0$ following partial differential equations hold:

\[ \sum_{1\leq j \leq n} \partial^{a}_{x_j} \partial^{b}_{y_j} f(X,Y) = 0. \]We have that $\symS_n$ act diagonally by permuting the variables in each alphabet, making the space of diagonal harmonics into a bigraded $\symS_n$-module $\mathcal{DH}_n.$ One can show that $\mathcal{DH}_n$ is also the ring of diagonal coinvariants

\[ \mathcal{DH}_n = \setC[X,Y]/I, \quad I = (\xvec,\yvec) \cap \setC[X,Y]^{\symS_n}. \]A result by M. Haiman [Hai94] states that the bigraded Frobenius characteristic of $\mathcal{DH}_n$ is given by $\nabla \elementaryE_n.$ The Shuffle theorem gives a combinatorial expression for $\nabla \elementaryE_n.$

### Definitions

Given a partition $\mu,$ we define

\[ B(\mu) \coloneqq \sum_{\square \in \mu} q^{\arm'(\square)} t^{\leg'(\square)}. \]Thus, a cell in row $i,$ column $j$ contributes with $q^{j-1} t^{i-1}.$

**Example (Example $B(331)$).**

We have that $B(\mu) = 1+q+q^2+q^3 + t(1+q+q^2+q^3)+t^2,$ as the monomials arise from the following Young diagram:

$1$ | $q$ | $q^2$ | $q^3$ |

$t$ | $tq$ | $q^2t$ | $q^3t$ |

$t^2$ |

Let $f$ be any symmetric function. We define an operators $\Delta_f$ and $\Delta'_f$ on $\spaceSym(q,t),$ by setting

\[ \Delta_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) ] \cdot \macdonaldH_\mu \text{ and } \Delta'_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) - 1]\cdot \macdonaldH_\mu. \]Note the use of plethystic substitution. Hence, the modified Macdonald polynomials are by definition eigenfunctions of $\Delta_f.$ Moreover, we define $\nabla$ as the special case

\[ \nabla( \macdonaldH_\mu ) \coloneqq \Delta_{\elementaryE_n} = \prod_{(i,j) \in \mu} ( q^{i-1} t^{j-1} ) \cdot \macdonaldH_\mu. \]In other words,

\[ \nabla( \macdonaldH_\mu ) = q^{\partitionN(\mu)} t^{\partitionN(\mu')} \macdonaldH_\mu \]### Relations among the operators

In, [HRW18] it is stated that

\[ \Delta_{\elementaryE_k} \elementaryE_n = \Delta'_{\elementaryE_k} \elementaryE_n + \Delta'_{\elementaryE_{k-1}} \elementaryE_n. \]## $\mathcal{DH}_n$ and the shuffle theorem

### Properties of $\mathcal{DH}_n$

Let $DH_n(q,t)$ be the bigraded Frobenius characteristic of $\mathcal{DH}_n.$ As shown by M. Haimain [Hai94], $DH_n(q,t) = \nabla \elementaryE_n.$

The (total) dimension of $\mathcal{DH}_n$ is $(n+1)^{n-1}.$
This is sequence A000272 and in particular,
is the number of *parking functions* of size $n.$
The dimension can be refined by degree —
the bigraded Hilbert series is given by

It is an open problem to give a combinatorial argument why this sum is symmetric in $q$ and $t.$

It is a major open problem to find a combinatorial expression for the Schur expansion of $DH_n(q,t)$ (see [Prob. 2.15, Hag07]). However, a combinatorial formula for the LLT polynomials would solve this problem as well.

**Example (Example of $\nabla \elementaryE_n$ in the Schur basis).**

For $n=1,2,3$ we have the following Schur expansions

$n$ | $\nabla \elementaryE_n$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$1$ | $\schurS_{1}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$2$ | $\schurS_{2} + (q+t) \schurS_{11}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$3$ | $\schurS_{3}+(q^2+t^2+qt+q+t) \schurS_{21} + (q^3+t^3+q^2 t + qt^2 + qt) \schurS_{111}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

The latter formula was originally conjectured by R. Stanley.

Also, by summing over spanning trees of $\{0,1,\dotsc,n\},$ and weighting them by inversions, we have

\[ DH_n(q,1) = \sum_{T} q^{\inv(T)}. \]The $qt$-Catalan numbers show up here by computing the sign representation:

\[ \langle DH_n(q,t), \elementaryE_{n} \rangle = \sum_{ P \in \DP(n) } q^{\dinv(P)} t^{\area(P)} = \sum_{ P \in \DP(n) } q^{\area(P)} t^{\bounce(P)}. \]These identities were proved in [Thm. I.2, GH02].

M. Haiman proved in [Thm. 3.10, Hai02b] a connection with Macdonald polynomials, among many other things.

**Theorem (Haiman (2002)).**

Set $T_\mu \coloneqq t^{n(\mu)} q^{n(\mu')},$ $M\coloneqq(1-q)(1-t),$ and

\[ w_\mu \coloneqq \prod_{\square \in \mu} (q^{\arm(\square)} - t^{\leg(\square)+1}) (t^{\leg(\square)}- q^{\arm(\square)+1} ), \qquad \Pi_\mu \coloneqq \prod_{\square \in \mu/(1)} \left(1- q^{\arm'(\square)} t^{\leg'(\square)} \right). \]Then, with $\macdonaldH_\mu$ being the modified Macdonald polynomials,

\[ DH_n(q,t) = \sum_{\mu \vdash n} \frac{ T_\mu M \macdonaldH_\mu \Pi_\mu \elementaryE_n[B(\mu)] }{ w_\mu }. \]### The shuffle theorem

The shuffle conjecture was originally stated in [HHLRU05]. It was later refined to what is known as the compositional shuffle conjecture, presented in [HMZ12]. Both these conjectures have now been settled.

**Theorem (Carlsson–Mellit [CM17]).**

We have the identity

\[ \nabla \elementaryE_n = \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \]where the sum is taken over so-called word parking functions.

After using the zeta map, as defined in [p. 82, Hag07], the right hand side can be expressed using vertical-strip LLT polynomials as

\[ \nabla \elementaryE_n = \sum_{\avec \in \DP(n)} t^{\bounce(\avec)} \LLT_{\avec,\svec}(\xvec;q) \]where $\svec$ marks every corner as strict, see [AP18] for definitions.

In fact, they prove a stronger statement, the compositional shuffle theorem.

**Theorem (Carlsson–Mellit [CM17]).**

Let $C_a$ be the operator defined as

\[ C_a f[X] \coloneqq [z^a] (-1/q)^{a-1} f[X-(1-q^{-1})/z] \sum_{z\geq 0} z^m \completeH_m[X]. \]We then have the identity

\[ \nabla (C_{\alpha_1} C_{\alpha_2}\dotsm C_{\alpha_\ell} 1) = \sum_{\substack{w \in WPF(n) \\ touch(w)=\alpha }} t^{\area(F)}q^{\dinv(F)} \xvec_w \]where the sum is taken over word parking functions with $\alpha$ specifying the touch-points.

For more background and references on the Shuffle theorem see S. van Willigenburg's survey, http://de.arxiv.org/pdf/1905.06970.pdf.

### A generalized shuffle theorem

A generalization of the Shuffle theorem is given in [BHMPS21a], where on the combinatorial side, the sum now ranges over lattice path under any line (not just a diagonal).

### qt-Narayana numbers

The $qt$-Narayana numbers are described in [ADDHB14]. They may be defined as

\[ N_{a,b}(q,t) = \sum_{P \in Polyo(b,a+1-b)} q^{\area(P)}t^{\bounce(P)} \]where the sum is taken over certain pairs of non-intersecting paths confined to a $b \times (a+1-b)$-rectangle (so called parallelogram polyominoes).

The authors show in [Thm. 6.1, ADDHB14] that

\[ N_{a,b}(q,t) = (qt)^{a} \langle \nabla \elementaryE_{a-1}, \completeH_{b-1} \completeH_{a-b} \rangle. \]## Rational Shuffle theorem

A bigraded deformation of the $q,t$-Catalan numbers are introduced in [GM13]. The rational shuffle conjecture is then stated in [Eq. 51, GN15] and independently in [Hik14] and [Arm12].

The extended rational shuffle conjecture is first stated in [BGLX15]. They introduce certain operators $Q_{m,n},$ generalizing operators $\widetilde{P}_{m,n}$ by Gorsky and Mazin (which required that $m,n$ be comprime), and conjecture a combinatorial formula for the operators when applied to $(-1)^n.$

Note that $Q_{n+1,n} (-1)^n = \nabla \elementaryE_n,$ so we have the shuffle theorem as a special case.

Using notation similar to that of [Eq. (11.97), Hag15], and [QR18], we set, for coprime $m,n,$

\[ H_{(m,n)(\xvec;q,t)} \coloneqq \sum_{(m,n) \text{ parking functions } P} q^{\area(P)} t^{\dinv(P)} \gessel_{n,\DES(P)}(\xvec). \]The functions $H_{(m,n)}(\xvec;q,t)$ are also known as Hikita polynomials (for coprime $m,n$), after the introduction in [Hik14]. Bergeron, Garsia, Leven and Xin then introduce the extended Hikita polynomials,

\[ H_{(km,kn)(\xvec;q,t)} \coloneqq \sum_{(km,kn) \text{ parking functions } P} [ret(P)]_{1/t} q^{\area(P)} t^{\dinv(P)} \gessel_{n,\DES(P)}(\xvec). \]where $ret(P)$ is the smallest integer $j \gt 0$ such that the supporting path of $P$ goes through $(jm,jn).$ One can then show (see [Eq. 6.3, BGLX15]) that $H_{(m,n)}$ is a positive linear combination of vertical-strip LLT polynomials. In [KK18], a close relationship between Hikita polynomials and Catalan symmetric functions is studied.

The operators $Q_{m,n}$ are defined recursively via the Bergeron–Garsia operators $D_k$ (acting on $F \in \spaceSym$)

\[ D_k F[X] \coloneqq [z]^{k} F\left[ X + \frac{(1-q)(1-t)}{z} \right] \sum_{j \geq 0} (-z)^j \elementaryE_j[X]. \]The construction of the $Q_{m,n}$ is quite involved. Moreover, it seems that different authors use slightly different definitions which leads to a sign difference.

It is shown by [GN15] that for coprime $m,n,$ and in [QR18] that

\[ \nabla Q_{m,n} \nabla^{-1} = Q_{m+n,n} \qquad \nabla Q_{kn,n} \nabla^{-1} = Q_{(k+1)n,n}. \]A. Mellit proved the following theorem in [Mel16].

**Theorem (The (extended) rational shuffle theorem).**

For coprime $m,n$ and $k \ge 0,$ we have that

\[ H_{(km,kn)}(\xvec;q,t) = Q_{km,kn} (-1)^n. \]In fact, he proves the compositional refinement stated in [Conj. 3.3, BGLX15].

In [Hik14], it is shown that $B_{m,n}(\xvec;q,t)$ is the Frobenius character of the action in the homology of a certain Springer fiber in the affine flag variety equipped with a certain filtration.

The rational Catalan numbers are defined as $\catalan_{a/b} = \frac{1}{a+b}\binom{a+b}{a,b}$ for coprime $a,b.$ We have that for coprime $m,n$ that

\[ \langle H_{(m,n)}(\xvec;q,t), \elementaryE_n \rangle_* = \catalan_{m/n}. \]where $\langle \cdot, \cdot \rangle_*$ is a certain deformation of the Hall innter product, see [Eq.(4.6) and (6.29), BGLX15].

An expression for $H_{(m,n)}(\xvec;q,t)$ can also be given by using so called Tesler matrices.

See [QR18] for recent results on Schur coefficients of $H_{(m,n)}(\xvec;q,t).$

## The square paths theorem, $\nabla \powerSum_n$

The square path conjecture was formulated by N. Loehr and G. Warrington [LW07b]. It provides a combinatorial expression for $\nabla \powerSum_n$ as a sum over preference function.

A preference functions is a map $f:[n] \to [n].$ A parking function is any preference function such that $|f^{-1}([k])| \geq k$ for all $k \in [n].$ With the parking language, we think of $f(i)$ as the preference of car $i.$ There is a way to associate a lattice path to any preference function as follows. In an $n\times n$-square, write the numbers $f^{-1}(i)$ in column $i,$ starting from $i=1.$ The numbers in each column are increasing upwards, and the $j$th number written is placed in row $j.$ Then there is a unique lattice path from $(0,0)$ to $(n,n)$ using North and East steps, such that the numbers are written immediately to the right of the North steps.

**Example (Path from preference function).**

For example, we have that the preference function $(1,5,1,2,1)$ give rise to the path $P =\mathtt{nnneneene},$ and the diagram

$2$ | ||||

$4$ | ||||

$5$ | ||||

$3$ | ||||

$1$ |

The area of a preference function is the number of full cells above the lowest diagonal which contains a car. Thus, if the path is a Dyck path (and the preference function is a parking function), the area coincide with the classical notion of area. The dinv statistic is similar to the inv-statistic for LLT polynomials, but one must add the number of cars strictly below the $x=y$ diagonal.

E. Sergel [Ser17] gave a proof of the square path conjecture.

**Theorem (E. Sergel, (2017)).**

We have the expansion in the fundamental quasisymmetric functions

\[ (-1)^{n-1} \nabla \powerSum_n = \sum_{w \in Pref(n)} t^{\area(w)} q^{\dinv(w)} \gessel_{\mathrm{IDES}(w)}(\xvec), \]where the sum is taken over all preference functions of size $n.$

It is rather straightforward to express this as a sum over vertical-strip LLT polynomials. A path arising from a preference function of size $n,$ can be described using a weak composition $\alpha,$ such that $\alpha_i \coloneqq |f^{-1}(i)|.$ It is then rather straightforward how to define $\area(\alpha)$ and $\mathrm{carsBelow}(\alpha),$ the number of cars below the main diagonal. Finally, we can associate an $n$-tuple of skew shapes, $\nuvec,$ determined by the positions of the cars. We therefore get the following:

**Proposition.**

We have that

\begin{equation*} (-1)^{n-1} \nabla \powerSum_n = \sum{\alpha} t^{\area(\alpha)} q^{\mathrm{carsBelow}(\alpha)} \LLT_{\nuvec(\alpha)}(\xvec;q) \end{equation*}where the sum ranges over all weak compositions of length $n$ and size $n.$

**Example (Schur-expansion of $(-1)^{n-1} \nabla \powerSum_n$).**

$n$ | $(-1)^{n-1} \nabla \powerSum_n$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$1$ | $\schurS_{1}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$2$ | $\schurS_{2} + (q + t + q t) \schurS_{11}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$3$ | $\schurS_{3} + (q + q^2 + t + q t + q^2 t + t^2 + q t^2 + q^2 t^2)\schurS_{21} +$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$+(q^3 + q t + q^2 t + q^3 t + q t^2 + q^2 t^2 + q^3 t^2 + t^3 + q t^3 + q^2 t^3) \schurS_{111}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

## The Delta conjecture, $\Delta'_{\elementaryE_k} \elementaryE_n$

The Delta conjecture generalizes the Shuffle theorem, and was stated by J. Haglund, J. Remmel, A. Wilson [HRW18].

There are two versions of the Delta conjecture, the rise version and the valley version:

**Conjecture (Delta conjecture, J. Haglund, J. Remmel, A. Wilson).**

The rise version:

\[ \Delta'_{\elementaryE_k} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Rise(w)} \left(1 + z/t^{a_i(w)}\right) \]The valley version:

\[ \Delta'_{\elementaryE_k} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Val(w)} \left(1 + z/t^{d_i(w)+1}\right) \]The rise version has a compositional generalization, and this has recently been proved in https://arxiv.org/pdf/2011.11467.pdf.

### The generalized Delta conjecture, $\Delta_{\completeH_m} \Delta'_{\elementaryE_{n-k-1}} \elementaryE_n$

Also Conj 4.2, 4.3 https://arxiv.org/pdf/2003.12048.pdf

### The extended Delta conjecture, $\Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n$

J. Haglund, J. Remmel, A. Wilson [HRW18] also formulated the extended Delta conjecture, which gives a combinatorial formula for

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n, \quad k \lt n. \]D. Qiu and A. Wilson [QW] gave the valley version of the extended Delta conjecture.

**Conjecture (Extended Delta conjecture, J. Haglund, J. Remmel, A. Wilson).**

The rise version:

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n,r)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Rise(w)} \left(1 + z/t^{a_i(w)}\right) \]Valley version:

\[ \Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n,r)} t^{\area(F)}q^{\dinv(F)} \xvec_w \prod_{i \in Val(w)} \left(1 + z/t^{d_i(w)+1}\right) \]The valley version has been proved true for $q=0$ or $t=0.$

See also [(6.10), Ber20a]. For some recent results see [DI].

## Generalized Delta square conjecture

The Generalized delta square conjecture (stated in [DIW]) is about a combinatorial interpretation of

\[ \frac{[n-k]_t}{[n]_t} \Delta_{\completeH_m} \Delta_{\elementaryE_{n-k}} \omega(\powerSum_n). \]Remark 3.14 shows that this (if the conjecture is true) can be expressed as a sum of LLT polynomials.

See conj. 4.5, 4.6: Valley version stated in [IW].

Some questions solved in [Rom22].

## Conjecture on $\nabla^m \schurS_\lambda$

N. Loehr and G. Warrington [LW10] conjectured a combinatorial expansion of $\nabla^m \schurS_\lambda.$

This conjecture (in fact, a much stronger result) was proved by J. Blasiak, M. Haiman, J. Morse, A. Pun and G. Seelinger, in [BHMPS21bBHMPS21c]. Their work also gives a new proof of the Shuffle theorem and similar other identities.

## Conjecture on $\nabla \monomial_\lambda$

**Conjecture (Bergeron–Garsia–Haiman–Tesler (1999), [Conj. IV, BGHT99]).**

We have for every pair of partitions $\lambda,$ $\mu,$

\[ (-1)^{|\lambda|-\length(\lambda)} \langle \nabla \monomial_\lambda, \schurS_\mu \rangle \in \setN[q,t]. \]This also appears in [Conj. 11, LW07b].

## Other conjectures

**Conjecture (Bergeron–Garsia–Haiman–Tesler (1999),[Conj. I, BGHT99]).**

We have for every pair of partitions $\lambda,$ $\mu$ and positive integer $m,$

\[ (-1)^{\iota(\lambda)} \langle \nabla^m \schurS_\lambda, \schurS_\mu \rangle \in \setN[q,t], \]where

\[ \iota(\lambda) \coloneqq \binom{\length(\lambda)}{2} + \sum_{\lambda_i < (i-1)} (i-1-\lambda_i). \]**Conjecture (Bergeron–Garsia–Haiman–Tesler (1999), [Conj. II, BGHT99]).**

We have for every pair of partitions $\lambda,$ $\mu,$

\[ (-1)^{|\lambda|-\length(\lambda)} \langle \nabla \macdonaldH_\lambda(\xvec;0,t), \schurS_\mu \rangle \in \setN[q,t], \]where

\[ \iota(\lambda) \coloneqq \binom{\length(\lambda)}{2} + \sum_{\lambda_i < (i-1)} (i-1-\lambda_i). \]In [BGHT99] and later [Conj. 3.20, Hai02b], the following conjecture is stated:

**Conjecture (Bergeron–Garsia–Haiman–Tesler (1999)).**

The expression

\[ \nabla_{\schurS_\nu} \elementaryE_n \]is $(q,t)$-Schur positive polynomial for all $\nu$ and $n.$

## List of papers and preprints

- Stephanie van Willigenberg, Overview of shuffle conjecture http://de.arxiv.org/pdf/1905.06970.pdf
- Nancy Wallace : https://arxiv.org/pdf/1906.08740.pdf
- Poster: http://fpsac2019.fmf.uni-lj.si/resources/Posters/115poster.pdf
- Zabrocki http://de.arxiv.org/pdf/1902.08966.pdf
- Wallach, on Zabrocki conjecture. http://de.arxiv.org/pdf/1906.11787.pdf
- Qiu, Wilson, Valley version of Delta conj, https://arxiv.org/pdf/1907.00268.pdf
- Haglund, Rhoades, Shimozono ORDERED SET PARTITIONS, GENERALIZED COINVARIANTALGEBRAS, AND THE DELTA CONJECTURE http://de.arxiv.org/pdf/1609.07575.pdf
- Shuffle conj and Hikita polynomials: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p21/pdf
- Natural extensions of parking space, https://arxiv.org/pdf/2003.04134.pdf
- Harmonic bases for generalized coinvariant algebras Brendon Rhoades, Tianyi Yu, Zehong Zhao, https://arxiv.org/abs/2004.00767
- Haglund and Sergel, New conjectures, http://de.arxiv.org/pdf/1908.04732.pdf
- Rational qt-Catalan https://arxiv.org/pdf/1908.11763.pdf
- Valley version of the Delta Square conjecture, https://arxiv.org/pdf/2003.12048.pdf
- F. Bergeron, Rational qt-catalan, lots of nice conjectures https://arxiv.org/pdf/1603.04476.pdf
- F. Bergeron, more e-positivity https://arxiv.org/pdf/1909.03531.pdf and https://arxiv.org/pdf/2003.07402.pdf
- Multialphabet bosonic-fermionic conjecture: https://arxiv.org/pdf/2005.00924.pdf.
- Gillespie, Rhoades: https://arxiv.org/pdf/2005.02110.pdf

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