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].
E. Cainelli, L. Luccioli, A. Iraci, and G. Paolini introduce \(qt\)Bench, a benchmark of 29 open \(q,t\)-combinatorics problems and one solved calibration problem, with finite data and automated checks [CLIP26]. Several problems concern Theta, Delta, \(\Xi,\) super-nabla, or path and parking-function statistics. The proposed area–bounce bijection remains under assessment, while negative coefficients rule out statistic interpretations for three other targets.
#Introduction
The ordinary coinvariant algebra uses one set of variables; here we use two sets with a simultaneous permutation action.
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 [Hai94, Sec. 1.2] and [Hag07, Sec. 2].
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:
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.
#Theta and Neguț operators
Theta operators are degree-raising operators on symmetric functions that package decorated refinements of Delta-operator expressions. In particular, the operators \(\Theta_{\elementaryE_k}\) record decorations on rises and valleys in several extensions of the Delta conjecture. The rational shuffle theorem uses a second family, the Neguț operators \(D_\gamma,\) indexed by integer sequences \(\gamma\); their action on the constant function \(1\) produces the corresponding Catalanimal symmetric functions.
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 \(t=1.\)
At \(t=1,\) the power-sum Theta operators on positive-degree symmetric functions are commuting derivations. Their action on a single-row modified Macdonald polynomial is identified with a normalized Macdonald cumulant, which in turn has a positive expansion in vertical-strip LLT polynomials [HT26, Thms. 1.2–1.3].
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.
M. D’Adderio, G. Interdonato, A. Iraci, and R. Pagaria give an explicit realization of the Neguț operators \(D_\gamma\) inside the Carlsson–Mellit Dyck path algebra \(\mathbb{A}_{q,t}\) [DIIP26, Thms. 4.11 and 4.20]. Their formula computes \(D_\gamma F\) for an arbitrary symmetric function \(F\) with a number of operator steps linear in the length of \(\gamma,\) and recovers the rational shuffle theorem without passing through the elliptic Hall algebra.
They also extend the Theta operators to an endomorphism of \(\mathbb{A}_{q,t}\) and derive explicit commutation rules with \(D_\gamma\) [DIIP26, Thms. 5.1, 5.12, and 5.14]. As an application, they prove the Theta conjecture at \(q=1,\) giving the rise-and-valley-decorated Dyck-path formula for \(\Theta_{\elementaryE_k}\Theta_{\elementaryE_\ell} \nabla \elementaryE_{n-k-\ell}\) [DIIP26, Thm. 6.1].
#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 \(\mathcal{DH}_n\)
#Hilbert and Frobenius series
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 [Hag07, Prob. 2.15] and [Wil19, Sec. 3]. A combinatorial Schur expansion for the relevant vertical-strip LLT polynomials would solve this problem as well.
#Bases and superspace coinvariants
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].
For the diagonal superspace coinvariant ring \(R_n^{(2,1)},\) N. González, J. Lentfer, and H. Mularczyk identify the triply graded sign component with that of \(R_n^{(2,0)}\otimes R_n^{(0,1)}.\) Its Hilbert series is the little Schröder polynomial, proving the sign-character case of Zabrocki’s conjecture [GLM26, Thms. 4.20–4.21 and Cors. 4.23–4.26].
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].
#Examples and specializations
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 [GH02, Thm. I.2].
M. Haiman proved the following Macdonald polynomial formula, among many other results [Hai02, Thm. 3.10].
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 [Hag07, p. 82], 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 [ADDH+14, Thm. 6.1] 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, Hikita polynomials, square paths, and Delta-type formulas are collected on the shuffle and Delta conjectures page.
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