Diagonal harmonics

2026-08-17 Improve this page

For background, see the canonical sources by J. Haglund [Hag07], and F. Bergeron [Ber09]. There has been substantial development since the writing of these books. B. Rhoades surveys symmetric-function identities tied to Delta operators and their representation-theoretic and geometric realizations [Rho22]. A. Garsia, J. Haglund, and M. Romero introduce several symmetric-function tools for Delta operators in [GHR19]. M. D'Adderio, A. Iraci, and A. V. Wyngaerd prove decorated Dyck-path and parallelogram-polyomino formulas for important Delta-conjecture scalar products [DIW22]. For generalized coinvariant algebras and ordered set partitions related to the Delta conjecture, see [HRS18, Gri21].

#Introduction

Let $X=(x_1,\dotsc,x_n)$ and $Y=(y_1,\dotsc,y_n).$ The symmetric group $\symS_n$ acts diagonally on $\setC[X,Y]$ by \[\sigma(x_i)=x_{\sigma(i)}, \qquad \sigma(y_i)=y_{\sigma(i)}.\] Let $\setC[X,Y]^{\symS_n}_{+}$ be the vector space of diagonal invariant polynomials with zero constant term, and let \[I_n \coloneqq \left\langle \setC[X,Y]^{\symS_n}_{+}\right\rangle.\] The diagonal coinvariant ring is \[\mathrm{DR}_n \coloneqq \setC[X,Y]/I_n.\] Equivalently, $I_n$ is generated by the polarized power sums \[p_{a,b}(X,Y)\coloneqq \sum_{i=1}^{n}x_i^a y_i^b, \qquad a,b\geq 0,\quad a+b\gt 0,\] see [Sec. 1.2, Hai94] and [Sec. 2, Hag07].

The dual differential-operator picture is the space of diagonal harmonics \[\mathcal{DH}_n \coloneqq \left\{ f\in\setC[X,Y]: \sum_{i=1}^{n} \partial_{x_i}^{a}\partial_{y_i}^{b}f=0 \text{ for all } a,b\geq0 \text{ with } a+b\gt0 \right\}.\] The diagonal action makes both $\mathrm{DR}_n$ and $\mathcal{DH}_n$ into bigraded $\symS_n$-modules. Under the usual apolar pairing, the two modules have the same bigraded Frobenius characteristic.

M. Haiman proved that \[\mathrm{Frob}_{q,t}(\mathcal{DH}_n)=\nabla \elementaryE_n, \qquad \dim \mathcal{DH}_n=(n+1)^{n-1},\] settling conjectures of Garsia and Haiman [Hai94, Hai02]. The Shuffle theorem gives a combinatorial expression for $\nabla\elementaryE_n,$ where $\nabla$ is the Macdonald eigenoperator defined below.

Example (The case $n=2$).

For $n=2,$ the harmonic space has basis \[1,\qquad x_2-x_1,\qquad y_2-y_1.\] The constant vector spans the trivial representation, while $x_2-x_1$ and $y_2-y_1$ span two copies of the sign representation in bidegrees $(1,0)$ and $(0,1).$ Thus \[\mathrm{Frob}_{q,t}(\mathcal{DH}_2) = \schurS_2 + (q+t)\schurS_{11},\] which agrees with $\nabla\elementaryE_2.$

#Macdonald eigenoperators

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 $B(441)$).

We have \[B(\mu) = 1+q+q^2+q^3 + t(1+q+q^2+q^3)+t^2,\] 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. The operators $\Delta_f$ and $\Delta'_f$ on $\spaceSym(q,t)$ are defined diagonally on the modified Macdonald basis by \[\Delta_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) ] \cdot \macdonaldH_\mu \text{ and } \Delta'_f( \macdonaldH_\mu ) \coloneqq f[ B(\mu) - 1]\cdot \macdonaldH_\mu.\] Note the use of plethystic substitution. For $\mu\vdash n,$ the $\nabla$ operator is the special case $\nabla=\Delta_{\elementaryE_n},$ so \[\nabla( \macdonaldH_\mu ) = \elementaryE_n[B(\mu)]\macdonaldH_\mu = \prod_{(i,j) \in \mu} q^{j-1} t^{i-1}\macdonaldH_\mu.\] In other words, \[\nabla( \macdonaldH_\mu ) = q^{\partitionN(\mu')} t^{\partitionN(\mu)} \macdonaldH_\mu.\]

E. Carlsson and A. Mellit give a combinatorial formula for the nabla operator [CM25]. This supplies another explicit model for the Macdonald-eigenoperator that governs diagonal harmonics.

F. Bergeron, J. Haglund, A. Iraci, and M. Romero introduce the super nabla operator [BHIR23]. It is a generic Macdonald eigenoperator from which the usual nabla operator, Delta operators, and several other Macdonald eigenoperators are obtained by specialization.

A. Iraci and M. Romero give elementary symmetric function expansions for a large family of Delta and Theta operator expressions at $t=1$ [IR24]. Their formulas are indexed by $\gamma$-parking functions and lattice $\gamma$-parking functions, and they lead to further $e$-positivity conjectures away from the specialization $t=1.$

M. D'Adderio and M. Romero prove identities for Theta operators [DR20]. These identities clarify how the Theta operators interact with Delta operators and plethystic substitutions.

#Relations among the operators

For $1\leq k\leq n,$ the two Delta operators above are related by [HRW18] \[\Delta_{\elementaryE_k} \elementaryE_n = \Delta'_{\elementaryE_k} \elementaryE_n + \Delta'_{\elementaryE_{k-1}} \elementaryE_n.\]

#$\mathcal{DH}_n$ and the shuffle theorem

#Properties of $\mathcal{DH}_n$

Let $\mathcal{F}_{\mathcal{DH}_n}(q,t)$ be the bigraded Frobenius characteristic of $\mathcal{DH}_n.$ By Haiman’s theorem, \[\mathcal{F}_{\mathcal{DH}_n}(q,t)=\nabla \elementaryE_n.\]

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 \[\mathrm{Hilb}_{\mathcal{DH}_n}(q,t) \coloneqq \left\langle \mathcal{F}_{\mathcal{DH}_n}(q,t), \completeH_{1^n} \right\rangle.\] The Shuffle theorem gives the parking-function formula \[\mathrm{Hilb}_{\mathcal{DH}_n}(q,t) = \sum_{w \in PF(n)} q^{\dinv(w)} t^{\area(w)}.\] It is an open problem to give a combinatorial argument why this sum is symmetric in $q$ and $t.$

It is still a major open problem to find a manifestly Schur-positive combinatorial formula for the Schur expansion of $\mathcal{F}_{\mathcal{DH}_n}(q,t)$; see [Prob. 2.15, Hag07] and [Sec. 3, Wil19]. A combinatorial Schur expansion for the relevant vertical-strip LLT polynomials would solve this problem as well.

Y. Jiang constructs an explicit vector-space basis for the alternating component of the diagonal coinvariant ring in terms of bivariate Vandermonde determinants indexed by Dyck paths [Jia25]. The bidegrees of these basis elements recover the usual $q,t$-Catalan formula, and the paper also proposes a related basis for generalized diagonal coinvariants indexed by $m$-Dyck paths.

S. Murai, B. Rhoades, and A. Wilson prove the Fields conjectures for the superspace coinvariant ring [MRW25]. They identify its ungraded $\symS_n$-module structure with a sign-twisted permutation action on ordered set partitions, and compute the bigraded Frobenius characteristic.

Three 2026 extensions make this superspace picture more uniform. Y. Jiang and J. Lentfer compute the bigraded Frobenius characteristic of the type-$B$ $(0,2)$ fermionic coinvariant ring; every irreducible type-$B$ character occurs with a single Schur-polynomial multiplicity, giving a multiplicity-free $\GL_2\times B_n$-module [JL26]. S. Bhattacharya and B. Rhoades prove the Sagan–Swanson monomial-basis and inverse-system conjectures for superspace coinvariants of the wreath products $\setZ_r\wr\symS_n$ [BR26]. Finally, B. Rhoades and A. Wilson give bigraded Hilbert-series and inverse-system formulas for superspace coinvariants of $\GL_n(\mathbb F_q),$ extending them to subgroups containing $\mathrm{SL}_n(\mathbb F_q)$ [RW26].

J. Kim constructs a combinatorial basis for the fermionic diagonal coinvariant ring using noncrossing partitions [Kim23]. The result gives a combinatorial model for the full ring as an $\symS_{n-1}$-module, extending earlier descriptions of its maximal degree components. J. Kim and B. Rhoades study set partitions, fermions, and skein relations in this setting [KR22]. Their work gives presentations and bases for related fermionic modules indexed by set partitions.

A. Iraci, B. Rhoades, and M. Romero prove the fermionic Theta coinvariant conjecture in the purely fermionic setting [IRR23]. This confirms the Theta-operator prediction for the corresponding quadruply graded $\symS_n$-module after setting the commuting variables to zero.

N. Wallace gives explicit combinatorial formulas for some irreducible characters of the $\GL_k\times \symS_n$-module of multivariate diagonal harmonics [Wal19]. For related generalized coinvariant algebras, see B. Rhoades, T. Yu, and Z. Zhao on harmonic bases [RYZ20], M. Gillespie and B. Rhoades on higher Specht bases [GR21], and M. Konvalinka and V. Tewari on natural extensions of the parking space [KT21].

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} $

Some special values of the Hilbert series are \[\mathrm{Hilb}_{\mathcal{DH}_n}(q,0) = [n]_q!, \qquad q^{\binom{n}{2}}\mathrm{Hilb}_{\mathcal{DH}_n}(q,q^{-1}) = ([n+1]_q)^{n-1}.\] These formulas are part of the specialization picture around Haiman’s $n!$ theorem [Hai94, Hai02]; 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 \[\mathrm{Hilb}_{\mathcal{DH}_n}(q,1) = \sum_{T} q^{\inv(T)}.\]

The $qt$-Catalan numbers appear here by computing the sign representation: \[\left\langle \mathcal{F}_{\mathcal{DH}_n}(q,t), \elementaryE_{n} \right\rangle = \sum_{ P \in \DP(n) } q^{\dinv(P)} t^{\area(P)} = \sum_{ P \in \DP(n) } q^{\area(P)} t^{\bounce(P)}.\] These identities were proved in [Thm. I.2, GH02].

M. Haiman proved the following Macdonald polynomial formula, among many other results [Thm. 3.10, Hai02].

Theorem (Haiman (2002)).

Set $T_\mu \coloneqq q^{\partitionN(\mu')}t^{\partitionN(\mu)},$ $M\coloneqq(1-q)(1-t),$ 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, \[\mathcal{F}_{\mathcal{DH}_n}(q,t) = \sum_{\mu \vdash n} \frac{ T_\mu M \Pi_\mu B(\mu)}{ w_\mu } \macdonaldH_\mu .\]

#The shuffle theorem

The shuffle conjecture was originally stated in [HHLR+05]. 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 \[\nabla \elementaryE_n = \sum_{\pi \in WPF(n)} t^{\area(\pi)}q^{\dinv(\pi)} \gessel_{n,\mathrm{ides}(\pi)}(\xvec),\] where the sum is taken over word parking functions. The statistic $\mathrm{ides}(\pi)$ records the inverse descent set of the parking function reading word. The functions $\gessel_{n,S}$ are the Gessel fundamental quasisymmetric functions.

Applying 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.

Carlsson and Mellit prove the stronger compositional shuffle theorem.

Theorem (Carlsson–Mellit [CM17]).

Let $C_a$ be the operator defined as \[C_a f[X] \coloneqq [z^a] (-1/q)^{a-1} f[X-(1-q^{-1})/z] \sum_{m\geq 0} z^m \completeH_m[X].\] We then have the identity \[\nabla (C_{\alpha_1} C_{\alpha_2}\dotsm C_{\alpha_\ell} 1) = \sum_{\substack{\pi \in WPF(n) \\ \mathrm{touch}(\pi)=\alpha }} t^{\area(\pi)}q^{\dinv(\pi)} \gessel_{n,\mathrm{ides}(\pi)}(\xvec),\] where $\alpha$ specifies the touch composition.

For more background and references on the Shuffle theorem, see the survey by S. v. Willigenburg [Wil19].

#A generalized shuffle theorem

A generalization of the Shuffle theorem is given in [BHMP+21]. On the combinatorial side, the sum ranges over lattice paths under any line, not just a diagonal.

J. Blasiak, M. Haiman, J. Morse, A. Pun, and G. H. Seelinger prove a nonsymmetric shuffle theorem in [BHMP+25]. Their framework uses modified nonsymmetric Macdonald polynomials, a nonsymmetric nabla operator, and nonsymmetric analogues of flagged LLT polynomials, and it supplies a natural setting for atom-positivity questions around nonsymmetric diagonal harmonics.

#$qt$-Narayana numbers

The $qt$-Narayana numbers are described in [ADDH+14]. 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 nonintersecting paths confined to a $b \times (a+1-b)$-rectangle, called parallelogram polyominoes.

The authors show in [Thm. 6.1, ADDH+14] that \[N_{a,b}(q,t) = (qt)^{a} \langle \nabla \elementaryE_{a-1}, \completeH_{b-1} \completeH_{a-b} \rangle.\] The specialization at $q=t=1$ can also be seen directly from the Shuffle theorem.

Proposition (Narayana coefficient at $q=t=1$).

For $0\leq k\leq n,$ \[\left. \langle \nabla \elementaryE_n,\completeH_k\completeH_{n-k}\rangle \right|_{q=t=1} = \frac{1}{n+1}\binom{n+1}{k+1}\binom{n+1}{k}.\]

Proof

By the Shuffle theorem and Hall duality, the scalar product is the number of word parking functions of size $n$ with monomial $x_1^k x_2^{n-k}.$ With only the labels $1$ and $2,$ every column has height at most two. If the supporting Dyck path has $j$ columns of height two, then the number of such paths is the Motzkin-triangle number \[\frac{n!}{(n-2j)!j!(j+1)!}.\] The $j$ two-car columns must contain both labels, and among the remaining $n-2j$ one-car columns we choose $k-j$ columns to carry label $1.$ Thus the desired number is \[\sum_{j\geq 0} \frac{n!}{(n-2j)!j!(j+1)!}\binom{n-2j}{k-j}.\] Multiplying by $n+1,$ rewriting the summand, and applying Vandermonde gives \[\begin{aligned} (n+1)\sum_{j\geq 0} \frac{n!}{(n-2j)!j!(j+1)!}\binom{n-2j}{k-j} &= \binom{n+1}{k+1} \sum_{j\geq 0}\binom{k+1}{j+1}\binom{n-k}{j} \\ &= \binom{n+1}{k+1}\binom{n+1}{k}. \end{aligned}\]

E. Gorsky, M. Mazin, and M. Vazirani study recursions for rational $q,t$-Catalan numbers [GMV20]. F. Bergeron gives a broad list of open questions for operators connected with rectangular Catalan combinatorics [Ber17].

#Rational Shuffle theorem

A bigraded deformation of the $q,t$-Catalan numbers is 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 coprime), 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.

#Hikita polynomials

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]. F. Bergeron, A. Garsia, E. Leven, and G. 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 $(jm,jn).$ They show [Eq. (6.3), BGLX15] that $H_{(m,n)}$ is a positive linear combination of vertical-strip LLT polynomials. R. Kaliszewski and D. Karmakar study a close relationship between Hikita polynomials and Catalan symmetric functions [KK18].

Nicolle Gonzalez, Jose Simental, and M. Vazirani introduce higher-rank $(q,t)$-Catalan polynomials [GSV23]. These are finite-variable truncations of Hikita polynomials and are related to parabolic affine Springer fibers. Their finite rational shuffle theorem uses rank-$r$ semistandard parking functions, with a dinv statistic coming from standardization and a codinv statistic matching the dimensions of cells in an affine paving.

A. Iraci, R. Pagaria, G. Paolini, and A. V. Wyngaerd formulate rectangular analogues of the square paths conjecture and the rise version of the Delta conjecture [IPPW23]. They prove their rectangular paths conjecture when the side lengths of the rectangle are coprime.

A. Wilson defines symmetric functions $L_{M,N}$ as weighted sums over tuples of lattice paths and shows that they satisfy a recursion lifting the triply graded Khovanov–Rozansky homology of torus links [Wil23]. This gives the torus-link homology as a specialization of $L_{M,N}$ and relates the construction conjecturally to elliptic Hall algebra operators.

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 involved, and different normalizations in the literature can introduce sign differences.

E. Gorsky and A. Neguț prove the first identity below for coprime $m,n$ [GN15]; D. Qiu and J. Remmel prove the second [QR18]: \[\nabla Q_{m,n} \nabla^{-1} = Q_{m+n,n} \qquad \nabla Q_{kn,n} \nabla^{-1} = Q_{(k+1)n,n}.\]

V. Pons gives an explicit description of the zeta map on Dyck paths in terms of area sequences [Pon22]. This provides a direct and implementable way to see how the zeta map sends the pair of statistics $(\dinv,\area)$ to $(\area,\bounce).$

A. Mellit proved the following theorem [Mel16].

Theorem (The (extended) rational shuffle theorem).

For coprime $m,n$ and $k \geq 0,$ we have \[H_{(km,kn)}(\xvec;q,t) = Q_{km,kn} (-1)^n.\]

In fact, he proves the compositional refinement stated in [Conj. 3.3, BGLX15].

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

The rational Catalan numbers are defined as $\catalan_{a/b} = \frac{1}{a+b}\binom{a+b}{a,b}$ for coprime $a,b.$ For coprime $m,n,$ \[\langle H_{(m,n)}(\xvec;q,t), \elementaryE_n \rangle_* = \catalan_{m/n}.\] Here $\langle \cdot, \cdot \rangle_*$ is a deformation of the Hall inner product, see [Eq. (4.6) and Eq. (6.29), BGLX15].

There is also an expression for $H_{(m,n)}(\xvec;q,t)$ using Tesler matrices.

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

M. Gillespie, E. Gorsky, and S. T. Griffin show that the symmetric function in the Rise Delta theorem can be obtained from the Rational Shuffle theorem by applying a Schur skewing operator [GGG25]. Their combinatorial proof gives a new derivation of the Rise Delta theorem from the Rational Shuffle theorem.

#Tamari intervals and more sets of variables

N. Bergeron, C. Ceballos, and V. Pilaud connect Hopf algebras of pipe dreams with multivariate diagonal harmonics [Sec. 8, BCP18]. Their dominant pipe-dream Hopf algebra is related to $\nu$-Tamari lattices, and this leads to the notion of a Hopf chain: a nested tuple of Dyck paths $(\pi_1,\dotsc,\pi_r)$ with $\pi_1$ equal to the diagonal path and with every triple satisfying a compatible $\nu$-Tamari interval condition.

For $n\leq 4$ and any number $r$ of variable sets, they show that the $q,t$-Frobenius characteristic of the multivariate diagonal harmonic space $\mathcal{DH}_{n,r}$ is given by a sum over Hopf chains, weighted by a collar statistic and by LLT polynomials. When $r=3,$ the construction specializes to the Tamari-interval conjectural picture of F. Bergeron and Louis-Francois Preville-Ratelle, where the length of a longest chain in a Tamari interval is one of the expected statistics.

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

The square path conjecture was formulated by N. Loehr and G. Warrington [LW07]. It provides a combinatorial expression for $\nabla \powerSum_n$ as a sum over preference functions.

A preference function 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 $i.$ Associate a lattice path to a 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. See preference functions and square paths for a more detailed definition of the reading word and dinv statistic.

Example (Path from preference function).

For example, the preference function $(1,5,1,2,1)$ gives rise to the path $P=\mathtt{nnneneeene},$ 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 coincides 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 gave a proof of the square path conjecture [Ser17].

Theorem (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 $n.$

Since $\omega(\powerSum_n)=(-1)^{n-1}\powerSum_n,$ the left-hand side is $\nabla\omega(\powerSum_n).$ It is tempting to call some of the coefficients below type $B$ Narayana numbers, because the specialization at $q=t=1$ gives the classical type $B$ Narayana numbers $\binom{n}{k}^2.$ This should be understood only as an analogy: the finite-reflection-group type $B$ Catalan story has different combinatorics, involving shifted or reflection-group paths [Stu10, Lu19].

Proposition (Square-path Narayana coefficients at $q=t=1$).

For $0\leq k\leq n,$ \[\left. \langle \nabla\omega(\powerSum_n), \completeH_k\completeH_{n-k}\rangle \right|_{q=t=1} = \binom{n}{k}^2.\]

Proof

By Sergel’s theorem, the word version of the expansion, and Hall duality, the scalar product at $q=t=1$ counts word preference functions of size $n$ with monomial $x_1^kx_2^{n-k}.$ With only labels $1$ and $2,$ each of the $n$ columns is one of \[\emptyset,\qquad \{1\},\qquad \{2\},\qquad \{1,2\}.\] Suppose there are $j$ columns of type $\{1,2\}.$ Then there are $k-j$ columns of type $\{1\},$ $n-k-j$ columns of type $\{2\},$ and $j$ empty columns. Hence the number is \[\sum_{j\geq 0} \binom{n}{j}\binom{n-j}{n-2j}\binom{n-2j}{k-j}.\] This simplifies to \[\binom{n}{k}\sum_{j\geq0}\binom{k}{j}\binom{n-k}{j} =\binom{n}{k}^2\] by Vandermonde’s identity.

Proposition (Specialization at $t=1/q$).

Let \[S_{n,k}(q,t)\coloneqq \langle \nabla\omega(\powerSum_n), \completeH_k\completeH_{n-k}\rangle.\] Then \[q^{k(n-k)}S_{n,k}(q,q^{-1}) = \qbinom{n}{k}_q^2.\]

Proof

This is a coefficient extraction from [Thm. 4.30, DIW22]. Since $\nabla=\Delta_{\elementaryE_n}$ on homogeneous symmetric functions of degree $n,$ their theorem gives \[\left.\nabla\omega(\powerSum_n)\right|_{t=1/q} = q^{-\binom{n}{2}}\elementaryE_n[X[n]_q].\] The coefficient dual to $\completeH_k\completeH_{n-k}$ is the coefficient of the monomial $x_1^kx_2^{n-k}.$ Choosing the $k$ $q$-weights attached to $x_1$ and the $n-k$ $q$-weights attached to $x_2$ gives \[\left[x_1^kx_2^{n-k}\right] q^{-\binom{n}{2}}\elementaryE_n[X[n]_q] = q^{-k(n-k)}\qbinom{n}{k}_q^2.\]

The same result of D’Adderio–Iraci–Vanden Wyngaerd gives the sign-character specialization \[\left. \langle \nabla\omega(\powerSum_n),\elementaryE_n\rangle \right|_{t=1/q} = q^{-\binom{n}{2}}\frac{1}{1+q^n}\qbinom{2n}{n}_q,\] see [Prop. 8.3, DIW22].

This expression can also be written 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 $\alpha_i \coloneqq |f^{-1}(i)|.$ The path determines $\area(\alpha)$ 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. This gives the following:

Proposition

\[(-1)^{n-1} \nabla \powerSum_n = \sum_{\alpha} t^{\area(\alpha)} q^{\mathrm{carsBelow}(\alpha)} \LLT_{\nuvec(\alpha)}(\xvec;q)\] 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} $

E. Sergel’s proof uses schedules to enumerate preference functions with a fixed diagonal word. If $\tau$ has run lengths $\rho_k,\dotsc,\rho_0$ and $w^{(l)}(c)$ denotes the $l$-schedule number of a car $c,$ then [Thm. 2.1, Ser17] gives the product formula \[\sum_{\substack{\operatorname{diagword}(Pr)=\tau\\ l(Pr)=l}} t^{\area(Pr)}q^{\dinv(Pr)} = t^{\operatorname{maj}(\tau)} q^{\rho_0+\dotsb+\rho_{l-1}}\prod_{c=1}^n [w^{(l)}(c)]_q.\] The key shift step is that the multiset of $l$-schedule numbers differs from the ordinary parking-function schedule by replacing one run length: \[\{w^{(l)}(c):1\leq c\leq n\} = \{w_i:1\leq i\leq n\}\cup\{\rho_l\}\setminus\{\rho_0\}.\] This is [Thm. 3.1, Ser17], and its proof uses [Lem. 3.1, Ser17], a conjugate-partition multiset identity.

#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]. It has two versions, the rise version and the valley version. The rise version is now a theorem: it follows from the compositional Delta theorem of M. D'Adderio and A. Mellit [DM22], and also from the extended Delta theorem of J. Blasiak, M. Haiman, J. Morse, A. Pun, and G. H. Seelinger [BHMP+23].

Theorem (Rise Delta theorem).

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

The valley version remains open in full generality; in particular, symmetry of the combinatorial side is still part of the problem. J. Haglund and E. Sergel introduce schedule formulas for the valley side and use them to prove several special cases [HS20].

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

\[\Delta'_{\elementaryE_k} \elementaryE_n = [z^{n-k-1}] \sum_{w \in WPF(n)} t^{\area(w)}q^{\dinv(w)} \xvec_w \prod_{i \in Val(w)} \left(1 + z/t^{d_i(w)+1}\right)\]

The rise version has a compositional generalization, which is the result proved in [DM22].

D. Qiu and M. Zhang prove a nonsymmetric compositional Delta theorem using flagged LLT polynomials [QZ26]. Their signed and unsigned identities use nonsymmetric analogues of $\nabla$ and $\tau^*$; Weyl symmetrization recovers the ordinary compositional Delta theorem. They also formulate stable-atom positivity conjectures for the nonsymmetric identities.

A. Iraci and A. V. Wyngaerd use a pushing operation to pass from the valley Delta conjecture to the generalized valley Delta conjecture [IW22].

M. D'Adderio, A. Iraci, Y. L. Borgne, M. Romero, and A. V. Wyngaerd use Theta operators to give a symmetric-function formula enumerating tiered trees [DIBR+23]. They also formulate a general conjecture whose special cases are connected to unified forms of the Delta conjecture.

A. Lacabanne, P. Vaz, and A. Wilbert study two-row Delta Springer varieties, introduced by Griffin–Levinson–Woo to geometrically realize a symmetric function from the Delta conjecture [LVW25]. They describe the irreducible components combinatorially, compare them with ordinary and exotic Springer fibers, and extend the homology action to a degenerate affine Hecke algebra action. S. T. Griffin, J. Levinson, and A. Woo introduced these Delta Springer fibers and proved a $t=0$ Delta-conjecture formula through their geometry [GLW24].

A. Iraci, P. Nadeau, and A. V. Wyngaerd prove a unified Delta theorem at $t=0$ using segmented Smirnov words [INW23]. The symmetric function they interpret is related to a diagonal coinvariant ring with one set of commuting variables and two sets of anticommuting variables.

S. Corteel, Matthieu Josuat-Verges, and A. V. Wyngaerd prove a specialization of the Delta conjecture at $q=-1$ [CJW24]. The result gives a $t$-analogue of Euler numbers in terms of peaks and valleys of permutations.

M. D'Adderio and A. Iraci prove several consequences of the valley Delta conjectures [DI23], including the Schröder case of the valley Delta conjecture and the Schröder case of its square version.

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

The generalized valley Delta conjecture is equivalent to the original valley Delta conjecture, by work of A. Iraci and A. V. Wyngaerd [IW22]. Their work also relates the valley Delta conjecture to the corresponding valley Delta square conjectures.

#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 \[\Delta'_{\elementaryE_k} \Delta_{\completeH_r} \elementaryE_n, \quad k \lt n.\]

Theorem (Extended Delta theorem).

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

This was proved in [BHMP+23].

D. Qiu and A. Wilson [QW20] formulated the valley version of the extended Delta conjecture.

Conjecture (Valley extended Delta conjecture).

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

The valley version has been proved true for $q=0$ or $t=0.$

See also [Ber24]. For consequences and related special cases, see [DI23].

#Generalized Delta square conjecture

The generalized Delta square conjecture (stated in [DIW19]) 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 of [DIW19] shows that this, if the conjecture is true, can be expressed as a sum of LLT polynomials.

The full Delta square conjectures remain open in general. The valley version was formulated in [IW21], and the implications from the original valley Delta conjecture are discussed in [IW22].

Some related questions are solved in [Rom22].

A. Iraci, R. Pagaria, and G. Paolini prove a fall-decorated rational shuffle theorem [IPP25]. This gives a rational analogue of the fall Delta theorem and is designed to interact with the Delta square conjectures.

S. Corteel, A. Lazar, and A. V. Wyngaerd study decorated square-path enumerators at $q=-1$ [CLW24]. Using a cyclic cutting-and-pasting action, they show that the valley Delta square enumerator vanishes when $n-k$ is even, while in the odd case it becomes a positive polynomial related to $t$-Euler numbers.

J. Lentfer proposes a monomial basis for the $(1,2)$-bosonic–fermionic coinvariant ring [Len24]. The candidate basis interpolates between the Kim–Rhoades modified Motzkin-path basis and the super-Artin basis, has Zabrocki’s conjectured cardinality $2^{n-1}n!,$ and gives conjectural Hilbert and Frobenius series equivalent to the segmented-Smirnov-word formulas of Iraci–Nadeau–Vanden Wyngaerd. M. Zabrocki conjectures a module for the Delta conjecture [Zab19]; N. Wallach studies consequences of this conjectural module for the action of $\symS_n$ on polynomial differential forms [Wal19]. See also F. Bergeron’s bosonic–fermionic diagonal coinvariant modules conjecture [Ber20].

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

N. Loehr and G. Warrington 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. H. Seelinger, in [BHMP+25, BHMP+24]. Their work also gives a new proof of the Shuffle theorem and similar identities. D. Kim and J. Oh introduce Macdonald piece polynomials and obtain the Loehr–Warrington conjecture as a corollary of a more general framework [KO24]. F. Bergeron studies related $\GL_k\times\symS_n$-modules and nabla applied to hook-indexed Schur functions [Ber19].

#The monomial nabla conjecture

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, LW07].

D. Qiu and M. Zhang proved this conjecture in full generality, and in fact proved the following stronger statement.

Theorem (Qiu–Zhang [Thm. 1.2, QZ26]).

For all partitions $\alpha,\lambda\vdash n$ and all integers $r\geq 1,$ \[(-1)^{|\alpha|-\length(\alpha)} \left\langle \nabla^r \monomial_\alpha,\schurS_\lambda \right\rangle \in \setN[q,t].\]

Their proof expands $(-1)^{|\alpha|-\length(\alpha)}\monomial_\alpha$ positively in the functions $C_\beta(1),$ then applies the compositional shuffle theorems and Schur positivity of LLT polynomials. They also prove an $\elementaryE$-positive analogue after the substitution $q\mapsto q+1$ [Thm. 1.3, QZ26].

M. Qu and G. Xin prove Schur positivity of $(-1)^k\nabla \monomial_{2^k1^\ell}$ and give a parking-function interpretation [QX25]. Their proof establishes $C$-positivity for $(-1)^k\monomial_{2^k1^\ell}$ and then applies the compositional shuffle theorem.

#Other conjectures

The first Bergeron–Garsia–Haiman–Tesler conjecture in this group concerns signed Schur positivity of $\nabla^m\schurS_\lambda.$ This is now a theorem, following from the proof of the Loehr–Warrington conjecture discussed above.

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 \[\iota(\lambda) \coloneqq \binom{\length(\lambda)}{2} + \sum_{\lambda_i \lt (i-1)} (i-1-\lambda_i).\]

M. Qu proves the two-column cases of this conjecture and the adjacent Bergeron–Garsia–Haiman–Tesler dominance conjecture for modified Hall–Littlewood polynomials [Qu26].

In [BGHT99] and later [Conj. 3.20, Hai02], the following conjecture is stated:

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

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

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The symmetric functions catalog

#Introduction

#Macdonald eigenoperators

#Relations among the operators

#\(\mathcal{DH}_n\) and the shuffle theorem

#Properties of \(\mathcal{DH}_n\)

#The shuffle theorem

#A generalized shuffle theorem

#\(qt\)-Narayana numbers

#Rational Shuffle theorem

#Hikita polynomials

#Tamari intervals and more sets of variables

#The square paths theorem, \(\nabla \powerSum_n\)

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

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

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

#Generalized Delta square conjecture

#Conjecture on \(\nabla^m \schurS_\lambda\)

#The monomial nabla conjecture

#Other conjectures

Bibliography