The symmetric functions catalog

#Macdonald P polynomials

Macdonald polynomials, \(\{\macdonaldP_\lambda(\xvec;q,t) \}_{\lambda},\) were introduced by I. Macdonald in [Mac88]. They are a two-parameter extension of the Schur functions and unify the Jack polynomials and Hall–Littlewood polynomials. For column partitions one has \(\macdonaldP_{(1^r)}(\xvec;q,t)=\elementaryE_r(\xvec).\)

The master’s thesis by R. Langer [Lan09] gives an overview. L. Colmenarejo and A. Ram suggest the term bosonic Macdonald polynomials, while the nonsymmetric Macdonald polynomials are called electronic Macdonald polynomials, see [CR22]. L. Colmenarejo, C. F. Dunkl, and J.-G. Luque compare vector-valued and highest-weight Jack and Macdonald polynomials [CDL19]. Their results study when projections from vector-valued Jack or Macdonald polynomials commute with the relevant symmetric-group, Hecke-algebra, and Cherednik operator actions.

A. Guionnet and J. Huang study large-variable asymptotics of skew Macdonald and Jack polynomials [GH26]. Their results characterize logarithmic limits by variational problems, using a large-deviation principle for \(\theta\)-analogues of non-intersecting Bernoulli random walks. At \(\theta=1\) this recovers the lozenge-tiling variational picture of H. Cohn, R. Kenyon, and J. Propp.

The Koornwinder polynomials are the universal classical-type Macdonald family. Their symmetric, nonsymmetric, and relative versions, together with rhombic-staircase and compressed set-valued tableau formulas, are discussed on their own page [CMW24, CGR26]. G. Olshanski relates Macdonald polynomials to an extended Gelfand–Tsetlin graph [Ols21]. This puts Macdonald-positive kernels and coherent systems into the branching-graph framework. V. Gorin proves the corresponding law of large numbers for every normalized Macdonald-nonnegative specialization when \(0\leq q,t\lt{}1\) [Gor26, Thm. 1.1]. If \((\alpha_i)\) and \((\beta_j)\) are the parameters of the specialization and \(\lambda\) is sampled from its coherent measure on partitions of \(n,\) then \[\frac{\lambda_i}{n}\longrightarrow\alpha_i, \qquad \frac{\lambda'_j}{n}\longrightarrow\beta_j\] almost surely for every \(i,j.\) More strongly, the row and column frequencies converge uniformly with exponentially small deviation probability. This gives the probabilistic interpretation of the parameters that was previously known in the Schur, Jack, and Hall–Littlewood specializations. J. Oh studies cyclic sieving phenomena built from Macdonald polynomials [Oh22]. S. Koshida studies observables of periodic Macdonald processes using free field theory [Kos21]. R. Biswal, V. Chari, P. Shereen, and J. Wand define symmetric polynomials \(G_{\nu,\lambda}(\xvec;q)\) whose \(\lambda=0\) specialization is a specialized Macdonald polynomial and whose \(\nu=0\) specialization is the graded character of a level-two Demazure module for affine \(\mathfrak{sl}_{n+1}\) [BCSW19]. They prove Schur positivity under natural hypotheses and express the family through Macdonald polynomials. J. He constructs a characteristic map for the symmetric space of symplectic forms over a finite field [He20]. Under this map the spherical functions are sent to Macdonald polynomials with parameters \((q,q^2),\) and the construction gives Schur-positivity results for skew Macdonald polynomials at these parameters.

#The \((q,q^2)\) finite-field specialization

The specialization \(t=q^2\) is distinguished by He’s characteristic map for the symmetric space of symplectic forms over a finite field. If \(q\) is an odd prime power, the Schur coefficient \[\left\langle \macdonaldP_{\lambda/\mu}(\xvec;q,q^2),\schurS_\nu \right\rangle\] is nonnegative [He20, Thm. 0.1]. This is a numerical positivity statement at finite-field values of \(q,\) not a claim that the coefficients are polynomials in \(q\) with nonnegative integer coefficients.

For example, the degree-two polynomial is \[\macdonaldP_{(2)}(\xvec;q,t) = \schurS_{(2)}(\xvec) + \frac{q-t}{1-qt}\schurS_{(1,1)}(\xvec),\] and hence \[\macdonaldP_{(2)}(\xvec;q,q^2) = \schurS_{(2)}(\xvec) + \frac{q}{1+q+q^2}\schurS_{(1,1)}(\xvec),\] which is positive for positive real \(q,\) in particular for odd prime powers. The vector-valued Jack and Macdonald polynomials of [CDL19] live in a representation-valued setting; here they are recorded as a projection and highest-weight comparison rather than as a separate scalar symmetric-function basis.

#Creation formula and Macdonald characters

H. B. Dali and M. D’Adderio introduce an operator \(\Gamma\) whose positive modes successively create the integral-form Macdonald polynomial \(\macdonaldJ_\lambda\) from \(1\) [DD26, Thms. 1.1–1.2]. The formula exposes the dependence on \(\lambda\) through the alphabet \((q^{\lambda_1},q^{\lambda_2},\dotsc)\) and connects the Bergeron–Garsia–Haiman–Tesler operator viewpoint with shifted Macdonald polynomials.

The same construction defines two-parameter Macdonald characters. They are shifted symmetric functions and form a basis of the shifted symmetric functions [DD26, Thm. 1.5 and Cor. 4.4]; they are also characterized by their degree, vanishing on sufficiently small Young diagrams, and top homogeneous component [DD26, Thm. 4.5]. After the reparametrization \(q=1+\gamma\alpha\) and \(t=1+\gamma,\) their \(\gamma\to0\) limit recovers Jack characters [DD26, Prop. 5.4].

The associated positivity statements are conjectural. In particular, the paper proposes two-parameter extensions of the Goulden–Jackson Matchings–Jack conjecture and \(b\)-conjecture [DD26, Conjs. 7–8]; setting \(\gamma=0\) returns their Jack counterparts.

#Inner product characterization

Let \(\langle \cdot , \cdot \rangle_{q,t}\) denote the Macdonald inner product on symmetric functions such that \[\langle \powerSum_\lambda , \powerSum_\mu \rangle_{q,t} = z_\lambda \delta_{\lambda\mu} \prod_{i=1}^{\length(\lambda)} \frac{1-q^{\lambda_i}}{1-t^{\lambda_i}} .\] Note that at \(q=t=0,\) we recover the standard Hall inner product.

The Macdonald polynomials \(\macdonaldP_\lambda(\xvec;q,t)\) are defined as the unique family of polynomials such that \[\macdonaldP_\lambda(\xvec;q,t) = \monomial_\lambda(\xvec) + \sum_{\mu \prec \lambda} \eta_{\lambda\mu} \monomial_\mu(\xvec)\] and \(\langle \macdonaldP_\lambda, \macdonaldP_\mu \rangle_{q,t}=0\) whenever \(\lambda \neq \mu.\) Here, \(\prec\) denotes the dominance order on partitions.

This characterization is not efficient for actual computation. The tableau formula below is usually more practical.

#Eigenvector characterization

The Macdonald polynomials can also be characterized as eigenvectors to a certain Macdonald difference operator \(D,\) see [Mac88]. The operator \(D\) is given as \[D = \sum_{i=1}^n \prod_{j \neq i} \left( \frac{tx_i-x_j}{x_i-x_j} \right) T_{q,x_i}\] where \(T_{q,x_i} f(x_1,\dotsc,x_n) = f(x_1,\dotsc,x_{i-1},qx_i,x_{i+1},\dotsc,x_n).\)

Macdonald polynomials can also be produced via determinants; see [LLM98].

P. Schlösser studies intermediate Macdonald polynomials, which are invariant under a parabolic subgroup of the finite Weyl group and interpolate between nonsymmetric and symmetric Macdonald polynomials [Sch26]. The paper proves orthogonality, eigenfunction properties, norm formulas, and vector-valued interpretations using double-affine Hecke algebras.

L. Lapointe introduces \(m\)-symmetric functions, which are symmetric in the variables \(x_{m+1},x_{m+2},\dotsc\) but not necessarily in \(x_1,\dotsc,x_m\) [Lap22]. He constructs \(m\)-symmetric Macdonald and Schur functions and formulates positivity conjectures for the modified \(m\)-symmetric Macdonald polynomials. M. Concha and L. Lapointe establish the basic structure of the \(m\)-symmetric Macdonald polynomials, including orthogonality, norm, evaluation, inclusion, and Cauchy-type formulas [CL23]. B. Goodberry studies the same family as type \(A\) partially symmetric Macdonald polynomials and proves stability, an integral form, and Pieri-type rules [Goo25]. D. Orr and Milo B. Weising identify normalized fixed-point classes in parabolic flag Hilbert schemes with modified partially symmetric Macdonald polynomials [OW24]. This confirms the Goodberry–Orr conjectural geometric interpretation and gives an explicit formula for the induced involution on the polynomial representation.

An \(m\)-partition is a pair \(\Lambda=(\mathbf a;\lambda),\) where \(\mathbf a\) is a weak composition with \(m\) parts and \(\lambda\) is a partition. L. Lapointe and L. Pena give a tableau model for the \(m\)-symmetric Schur functions \(s_\Lambda(\xvec;0)\) [LP26]. Their restricted RSK correspondence proves the Cauchy identity \[\frac{1}{ \displaystyle \prod_{i+j\leq m+1}(1-x_i y_j) \prod_{i,j\geq1}(1-x_i y_j)} = \sum_\Lambda s_\Lambda(\xvec;0)s_\Lambda^*(\yvec).\] The dual functions are key polynomials. The proof uses a new global characterization of right keys.

The same authors prove the \(m\)-symmetric Macdonald positivity conjecture at \(t=1\) [LP26]. In particular, the coefficients in \[\lim_{t\to1} J_\Lambda\!\left[\frac{X}{1-t};q,t\right] = \sum_\Omega K_{\Omega\Lambda}(q,1)s_\Omega(\xvec;1)\] belong to \(\setN[q].\) When \(\Omega\) is dominant, they obtain the positive formula \[K_{\Omega\Lambda}(q,1) = \sum_{\operatorname{sh}(T)=\Omega} q^{\operatorname{maj}_{\widetilde{\Lambda}}(T_{\mathbf a})},\] over standard fillings of \(\Omega\); symmetry in the first \(m\) entries reduces the general case to this one.

M. Concha and L. Lapointe study bisymmetric Macdonald polynomials [CL23]. These are built from nonsymmetric Macdonald polynomials by combining antisymmetrization and \(t\)-symmetrization. They prove an evaluation symmetry extending the usual Macdonald symmetry, and derive Pieri rules for multiplication by elementary symmetric functions.

#Tableau formula

The classical tableau formula expresses Macdonald polynomials as a sum over tableaux; see [Mac88] and [Mac95]. We first need to introduce some notation. Given \(\lambda/\mu,\) let \(R_{\lambda/\mu}\) and \(C_{\lambda/\mu}\) be the set of rows and columns, respectively, containing some box of \(\lambda/\mu.\)

When \(\lambda/\mu\) is a horizontal strip (no two boxes in the same column), we define \(\psi_{\lambda/\mu}(q,t)\) as the following product. Set \[\psi_{\lambda/\mu}(q,t) \coloneqq \prod_{\substack{ s \in R_{\lambda/\mu} \setminus C_{\lambda/\mu} }} \frac{b_{\mu}(s;q,t)}{b_{\lambda}(s;q,t)}\] where \[b_{\mu}(s;q,t) \coloneqq \begin{cases} \frac{ 1-q^{\arm_\mu(s)}t^{1+\leg_\mu(s)} }{ 1-q^{1+\arm_\mu(s)}t^{\leg_\mu(s)} } &\text{if $s\in \mu$} \\ 1 &\text{otherwise}. \end{cases}\]

Now, if \(T\) is a semistandard Young tableau, we let \[\psi_{T}(q,t) \coloneqq \prod_{j=1}^n \psi_{\lambda^{j}/\lambda^{j-1}}(q,t)\] where \(\lambda^{j}/\lambda^{j-1}\) is the horizontal strip determined by the entries with value \(j\) in \(T.\)

The tableau formula is \[\macdonaldP_{\mu}(\xvec;q,t) = \sum_{\nu \vdash n} \monomial_\nu(\xvec) \sum_{T \in \SSYT(\mu,\nu)} \psi_{T}(q,t).\]

Note that if \(q=t,\) this formula gives the Schur polynomial \(\schurS_\mu(\xvec).\)

O. Mandelshtam gives a more compact tableau formula for the symmetric Macdonald polynomials \(\macdonaldP_\lambda(\xvec;q,t)\) using coquinv-sorted non-attacking tableaux and probabilistic operators [Man24]. The formula has fewer terms than the classical tableau formula and is connected by a bijection to multiline queues; in a limit, it recovers the Knop–Sahi formula for Jack polynomials.

E. Y. Jin and X. Lin use superization formulas for modified Macdonald polynomials to derive two formulas for \(\macdonaldP_\lambda(\xvec;q,t)\) [JL26, Thms. 1.3–1.4]. The first sums over non-attacking, top-row-increasing tableaux of the transposed diagram. Its compressed version sums over sorted non-attacking tableaux and includes a product of \(t\)-multinomial coefficients. When \(\lambda\) has repeated parts, the compressed formula has the fewest summands among the formulas compared in the paper. It also gives an explicit \((q,t)\)-formula for each monomial coefficient \([\monomial_\mu]\macdonaldP_\lambda.\)

#From nonsymmetric Macdonald \(E\) polynomials

J. Haglund, M. Haiman, and N. Loehr obtain the following expansion [HHL08, Prop. 5.3.1]: \[\macdonaldP_\lambda(\xvec;q,t) = \prod_{u \in \lambda} \left(1- q^{1+\leg(u)}t^{\arm(u)}\right) \sum_{\gamma \sim \lambda} \frac{ \macdonaldE_{\gamma}(x_1,\dotsc,x_n;q^{-1},t^{-1})}{ \prod_{v \in \gamma} \left(1- q^{1+\leg(v)}t^{\arm(v)}\right) }.\] The polynomial \(\macdonaldE_{\gamma}(\xvec;q,t)\) is a nonsymmetric Macdonald polynomial.

The same formula can also be expressed using permuted-basement Macdonald polynomials: \[\macdonaldP_\lambda(\xvec;q,t) = \prod_{u \in \lambda} \left(1- q^{1+\leg(u)}t^{\arm(u)}\right) \sum_{\gamma \sim \lambda} \frac{ \macdonaldE^{id}_{\gamma}(x_1,\dotsc,x_n;q,t)}{ \prod_{v \in \gamma} \left(1- q^{1+\leg(v)}t^{\arm(v)}\right) },\] since we have the relation \(\macdonaldE^{id}_{\gamma}(x_1,\dotsc,x_n;q,t) = \macdonaldE_{\gamma}(x_n,\dotsc,x_1;q^{-1},t^{-1}).\)

A related expression from [CR22] is \[\macdonaldP_\lambda(\xvec;q,t) = \frac{1}{W_\lambda(t)} \sum_{w \in \symS_n} w \left( \macdonaldE_{\gamma}(\xvec;q,t) \prod_{i \lt j} \frac{x_i - t x_j}{x_i - x_j} \right),\] where \(W_\lambda(t)\) is a normalization constant, so that \([x^{\lambda}] \macdonaldP_\lambda =1.\) Compare this with the formula for Hall–Littlewood polynomials, obtained as \(q\to 0.\)

In fact, an analogue of this formula exists for any fixed basement, see [Ale19, Thm. 29].

C. Lenart, S. Naito, F. Nomoto, and D. Sagaki give a path model for symmetric and nonsymmetric Macdonald polynomials of arbitrary type [LNNS25]. The model uses pseudo-quantum Lakshmibai–Seshadri paths and gives a counterpart of the Ram–Yip formula with less data than alcove walks. The paths also carry a connected pseudo-crystal structure.

#Specializations

The Macdonald polynomials specialize to other families of symmetric functions: \[\begin{aligned} \schurS_\lambda(\xvec) &= \macdonaldP_\lambda(\xvec;q,q), \\ \hallLittlewoodP_\lambda(\xvec;t) &= \macdonaldP_\lambda(\xvec;0,t), \\ \jackP_\lambda(\xvec;a) &= \lim_{t \to 1} \macdonaldP_\lambda(\xvec;t^a,t). \end{aligned}\]

#Macdonald J polynomials

By rescaling, the integral form Macdonald polynomials are defined via \[\macdonaldJ_\lambda(\xvec;q,t) = \left( \prod_{\square \in \lambda} 1-q^{\arm(\square)}t^{1+\leg(\square)} \right) \macdonaldP_\lambda(\xvec;q,t).\]

Similarly, the integral-form Jack polynomials are obtained as a limit: \[\jackJ_\lambda(\xvec;a) = \lim_{t \to 1} \frac{\macdonaldJ_\lambda(\xvec;t^a,t)}{(1-t)^{|\lambda|}}\]

The modified Macdonald polynomials (also known as transformed Macdonald polynomials) are obtained by letting \[\macdonaldH_\lambda(\xvec;q,t) \coloneqq t^{n(\lambda)}H_\lambda(\xvec;q,1/t), \qquad H_\lambda(\xvec;q,t) = \macdonaldJ_\lambda[\xvec/(1-t);q,t],\] and we use plethystic notation in the last expression.

#\(qt\)-Kostka polynomials

Macdonald [Mac88, p. 354] introduces the \(qt\)-Kostka polynomials \(K_{\lambda\mu}(q,t)\) via the relation \[\macdonaldJ_\mu(\xvec;q,t) = \sum_{\lambda} K_{\lambda\mu}(q,t) \schurS_\lambda[\xvec(1-t)]\] where \[\schurS_\lambda[\xvec(1-t)] = \sum_{\rho} \frac{\chi_\rho^\lambda}{z_\rho} \powerSum_\rho(x) \prod_{i=1}^{\length(\rho)} (1-t^{\rho_i}).\] Macdonald conjectured that the \(K_{\lambda\mu}(q,t)\) are polynomials in \(\setN[q,t]\) [Mac88]. M. Haiman later confirmed this [Hai01], as the property follows from the Schur positivity of the modified Macdonald polynomials, and the corresponding modified \(qt\)-Kostka polynomials.

It is still an open problem to find a combinatorial interpretation of \(K_{\lambda\mu}(q,t).\)

K. Matveev and Y. Wei prove a factorization formula for certain Macdonald Littlewood–Richardson coefficients \(c^\nu_{\lambda\mu}(q,t)\) [MW23]. Their result applies when the corresponding Kostka number \(K_{\mu,\nu-\lambda}\) is equal to \(1,\) giving a Macdonald analogue of a special case of a factorization conjecture of R. Stanley.

R. Mickler constructs a Macdonald Lax operator with an orthogonal eigenbasis indexed by partitions and addable corners, and derives a box-sum identity for Macdonald Littlewood–Richardson coefficients [Mic26]. A two-parameter strengthening, proved using Warnaar’s product formula, gives weighted linear relations among these coefficients.

#Haglund’s conjecture

J. Haglund formulates Haglund’s Macdonald conjecture, in [Hag10]: \[\left\langle \frac{\macdonaldJ_\lambda(\xvec;q,q^k)}{(1-q)^{|\lambda|}}, \schurS_\mu(\xvec) \right\rangle \in \setN[q]\] for any \(k \in 0,1,2,\dotsc.\) J. Haglund, M. Haiman, and N. Loehr prove that the expression \(\frac{\macdonaldJ_\lambda(\xvec;q,q^k)}{(1-q)^{|\lambda|}}\) has coefficients in \(\setN[q]\) when expanded in the \(\monomial_\mu\)-basis [HHL05]. Haglund’s conjecture states that the expression is actually Schur-positive.

A. Ram suggests that Haglund’s conjecture can be made stronger: a positivity statement should hold when \(\macdonaldJ_\lambda(\xvec;q,q^k)\) is expanded in the \(\macdonaldJ_\lambda(\xvec;q,q^{k-1})\)-basis. Note that \(\macdonaldJ_\lambda(\xvec;q,q)=\schurS_\lambda.\)

Haglund’s conjecture has been proved in some special cases; see [Yoo12, YOO15, Bha22].

M. Lassalle formulates conjectures for Macdonald polynomials of types \(B,\) \(C,\) and \(D\) [Las05].

#Quasisymmetric Macdonald J polynomials

A quasisymmetric refinement of \(\macdonaldJ_\lambda(\xvec;q,t)\) is introduced in [CHMM+22]. The quasisymmetric Macdonald polynomials \(G_{\alpha}(\xvec;q,t)\) are defined as sums over certain nonsymmetric Macdonald polynomials, such that the following properties hold:

• \(\macdonaldJ_\lambda(\xvec;q,t)\) is a positive sum of \(G_{\alpha}(\xvec;q,t).\)

• \(G_{\alpha}(\xvec;q,t)\) is quasisymmetric.

• \(G_{\alpha}(\xvec;0,0)\) is a quasisymmetric Schur function.

• The \(G_{\alpha}(\xvec;q,t)\) have a combinatorial HHL-type formula.

#Multi-Macdonald P polynomials

C. González and L. Lapointe introduce the multi-Macdonald polynomials, which are indexed by \(r\)-tuples of partitions [GL19]. They are defined via triangularity relations, similar to the definition of Macdonald P polynomials. When \(r=1,\) the usual Macdonald \(P\) polynomials are recovered, and when \(r=2,\) the family coincides with the double Macdonald polynomials, previously defined in [BLM14].

The multi-Macdonald polynomials can be expressed as a product of Macdonald \(P\) polynomials, with suitable plethystic substitutions in each factor.

The same construction has an analogue of the modified Macdonald polynomials, and then an analogue of the \(qt\)-Kostka polynomials when expanded in the Wreath product Schur polynomials. González and Lapointe show that the multi-\(qt\)-Kostka polynomials are in \(\setN[q,t].\)

#Wreath Macdonald P polynomials

The wreath Macdonald \(P\) polynomials, \(\macdonaldP_{\lambdavec^\bullet}(\xvec;q,t),\) were conjectured to exist in M. Haiman’s seminal article [Hai02, Sec. 7.2.19]. This conjecture was later proved in [BFV14]. See [OS23] for a survey on wreath Macdonald P polynomials.

J. J. Wen realizes wreath Macdonald polynomials as eigenstates in the vertex representation of a quantum toroidal algebra [Wen25]. In this model they diagonalize the horizontal Heisenberg subalgebra, and the proof also gives another construction of the wreath Macdonald polynomials using shuffle algebra methods.

D. Mathiä and U. Thiel show that wreath Macdonald polynomials specialized at \(q=t\) give characters of simple modules for restricted rational Cherednik algebras of wreath product groups [MT22]. This generalizes Gordon’s character formula for the symmetric group.

D. Orr and M. Shimozono give explicit \(q\)-difference operators acting diagonally on wreath Macdonald \(P\)-polynomials [OS24]. This gives a finite-variable operator model for the diagonalization property. D. Orr, M. Shimozono, and J. Wen construct difference-permutation operators diagonalized by wreath Macdonald \(P\)-polynomials [OSW25]. Their eigenvalues are elementary symmetric polynomials of arbitrary degree, and the operators come from integral formulas in the vertex representation of the corresponding quantum toroidal algebra.

M. Romero and J. J. Wen prove Tesler identities for wreath Macdonald polynomials [RW25]. Their operator formula sends a wreath Macdonald polynomial to a delta function attached to its character, leading to reciprocity results, wreath interpolation Macdonald polynomials, plethystic formulas for wreath \((q,t)\)-Kostka coefficients, and series solutions to the corresponding bispectral problem. S. A. Ferlinc and J. J. Wen introduce Ext-operator analogues of the Carlsson–Nekrasov–Okounkov vertex operator for wreath Macdonald polynomials [FW25]. As an application they prove a modular \((q,t)\)-Nekrasov–Okounkov formula for \(r\geq 3.\)

Bibliography

1. [Ale19]Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
   .bib

   Copy Download

   @article{Alexandersson2015gbMacdonald,
     author = {Per Alexandersson},
     title = {Non-symmetric {M}acdonald polynomials and {D}emazure--{L}usztig operators},
     journal = {Séminaire Lotharingien de Combinatoire},
     volume = {76},
     year = {2019},
     url = {https://www.mat.univie.ac.at/~slc/wpapers/s76alexand.html}
   }
   

2. [BFV14]Roman Bezrukavnikov, Michael Finkelberg and Vadim Vologodsky. Wreath Macdonald polynomials and the categorical McKay correspondence. Cambridge Journal of Mathematics, 2(2):163–190, 2014.
   .bib

   Copy Download

   @article{BezrukavnikovFinkelbergVologodsky2014,
     doi = {10.4310/cjm.2014.v2.n2.a1},
     url2 = {https://doi.org/10.4310/cjm.2014.v2.n2.a1},
     year = {2014},
     publisher = {International Press of Boston},
     volume = {2},
     number = {2},
     pages = {163--190},
     author = {Roman Bezrukavnikov and Michael Finkelberg and Vadim Vologodsky},
     title = {Wreath {M}acdonald polynomials and the categorical {McKay} correspondence},
     journal = {Cambridge Journal of Mathematics}
   }
   

3. [Bha22]Aritra Bhattacharya. Haglund’s positivity conjecture for multiplicity one pairs. arXiv:2205.11802, 2022.
   .bib

   Copy Download

   @article{Bhattacharya2022x,
   Author = {Aritra Bhattacharya},
   Title = {Haglund's positivity conjecture for multiplicity one pairs},
   Year = {2022},
   Eprint = {2205.11802},
     url = {https://arxiv.org/abs/2205.11802},
   journal = {arXiv e-prints}
   }
   

4. [BCSW19]Rekha Biswal, Vyjayanthi Chari, Peri Shereen and Jeffrey Wand. Macdonald Polynomials and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$. arXiv:1910.05848, 2019.
   .bib

   Copy Download

   @article{BiswalChariShereenWand2019x,
     author = {Rekha Biswal and Vyjayanthi Chari and Peri Shereen and Jeffrey
       Wand},
     title = {Macdonald {P}olynomials and level two {D}emazure modules for
       affine {$\mathfrak{sl}_{n+1}$}},
     year = {2019},
     eprint = {1910.05848},
     archivePrefix = {arXiv},
     primaryClass = {math.RT}
   }
   

5. [BLM14]O. Blondeau-Fournier, L. Lapointe and P. Mathieu. Double Macdonald polynomials as the stable limit of Macdonald superpolynomials. Journal of Algebraic Combinatorics, 41(2):397–459, July 2014.
   .bib

   Copy Download

   @article{BlondeauFournierLapointeMathieu2014,
     doi = {10.1007/s10801-014-0541-6},
     url2 = {https://doi.org/10.1007/s10801-014-0541-6},
     year = {2014},
     month = jul,
     publisher = {Springer Science and Business Media {LLC}},
     volume = {41},
     number = {2},
     pages = {397--459},
     author = {O. {Blondeau-Fournier} and L. Lapointe and P. Mathieu},
     title = {Double {M}acdonald polynomials as the stable limit of {M}acdonald superpolynomials},
     journal = {Journal of Algebraic Combinatorics}
   }
   

6. [CDL19]Laura Colmenarejo, Charles F. Dunkl and Jean-Gabriel Luque. Connections between vector-valued and highest weight Jack and Macdonald polynomials. arXiv:1907.04631, 2019.
   .bib

   Copy Download

   @article{ColmenarejoDunklLuque2019x,
     author = {Laura Colmenarejo and Charles F. Dunkl and Jean-Gabriel Luque},
     title = {Connections between vector-valued and highest weight {J}ack and
       {M}acdonald polynomials},
     year = {2019},
     eprint = {1907.04631},
     url = {https://arxiv.org/abs/1907.04631},
     journal = {arXiv e-prints}
   }
   

7. [CGR26]Laura Colmenarejo, Lucas Gagnon and Arun Ram. Formulas for Koornwinder polynomials. arXiv:2608.02810v1, 2026.
   .bib

   Copy Download

   @article{ColmenarejoGagnonRam2026x,
     author = {Laura Colmenarejo and Lucas Gagnon and Arun Ram},
     title = {Formulas for {K}oornwinder polynomials},
     year = {2026},
     eprint = {2608.02810v1},
     url = {https://arxiv.org/abs/2608.02810v1},
     journal = {arXiv e-prints}
   }
   

8. [CR22]Laura Colmenarejo and Arun Ram. C-functions and Macdonald polynomials. arXiv:2212.03312, 2022.
   .bib

   Copy Download

   @article{ColmenarejoRam2022x,
   Author = {Laura Colmenarejo and Arun Ram},
   Title = {c-functions and {M}acdonald polynomials},
   Year = {2022},
   Eprint = {2212.03312},
     url = {https://arxiv.org/abs/2212.03312},
   journal = {arXiv e-prints}
   }
   

9. [CL23]Manuel Concha and Luc Lapointe. The $m$-symmetric macdonald polynomials. arXiv:2311.12625, 2023.
   .bib

   Copy Download

   @article{ConchaLapointe2023Macdonald,
     author = {Manuel Concha and Luc Lapointe},
     title = {The {$m$}-symmetric Macdonald polynomials},
     year = {2023},
     eprint = {2311.12625},
     url = {https://arxiv.org/abs/2311.12625},
     journal = {arXiv e-prints}
   }
   

10. [CL23]Manuel Concha and Luc Lapointe. Symmetry and Pieri rules for the bisymmetric Macdonald polynomials. arXiv:2307.02385, 2023.
    .bib

    Copy Download

    @article{ConchaLapointe2023x,
      author = {Manuel Concha and Luc Lapointe},
      title = {Symmetry and {P}ieri rules for the bisymmetric {M}acdonald polynomials},
      year = {2023},
      eprint = {2307.02385},
      url = {https://arxiv.org/abs/2307.02385},
      journal = {arXiv e-prints}
    }
    

11. [CHMM+22]Sylvie Corteel, Jim Haglund, Olya Mandelshtam, Sarah Mason and Lauren Williams. Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials. Selecta Mathematica, 28(2), January 2022.
    .bib

    Copy Download

    @article{CorteelMandelshtamMasonWilliams2022,
      doi = {10.1007/s00029-021-00721-7},
      url = {https://doi.org/10.1007/s00029-021-00721-7},
      year = {2022},
      month = jan,
      publisher = {Springer Science and Business Media {LLC}},
      volume = {28},
      number = {2},
      author = {Sylvie Corteel and Jim Haglund and Olya Mandelshtam and Sarah Mason and Lauren Williams},
      title = {Compact formulas for {M}acdonald polynomials and quasisymmetric {M}acdonald polynomials},
      journal = {Selecta Mathematica}
    }
    

12. [CMW24]Sylvie Corteel, Olya Mandelshtam and Lauren Williams. Rhombic staircase tableaux and Koornwinder polynomials. Mathematische Zeitschrift, 308(3), 2024.
    .bib

    Copy Download

    @article{CorteelMandelshtamWilliams2024,
      author = {Corteel, Sylvie and Mandelshtam, Olya and Williams, Lauren},
      title = {Rhombic staircase tableaux and {K}oornwinder polynomials},
      year = {2024},
      journal = {Mathematische Zeitschrift},
      volume = {308},
      number = {3},
      publisher = {Springer Science and Business Media LLC},
      doi = {10.1007/s00209-024-03596-4},
      url = {http://dx.doi.org/10.1007/s00209-024-03596-4},
      issn = {1432-1823},
      eprint = {2312.17469}
    }
    

13. [DD26]Houcine Ben Dali and Michele D’Adderio. Macdonald characters from a new formula for Macdonald polynomials. Selecta Mathematica, 32(5), 2026.
    .bib

    Copy Download

    @article{DaliDAdderio2024x,
      author = {Houcine Ben Dali and Michele D'Adderio},
      title = {Macdonald characters from a new formula for {M}acdonald polynomials},
      year = {2026},
      journal = {Selecta Mathematica},
      volume = {32},
      number = {5},
      doi = {10.1007/s00029-026-01201-6},
      eprint = {2404.03904},
      url = {https://doi.org/10.1007/s00029-026-01201-6}
    }
    

14. [FW25]Seamus Albion Ferlinc and Joshua Jeishing Wen. Ext operators for wreath Macdonald polynomials. arXiv:2508.10772, 2025.
    .bib

    Copy Download

    @article{FerlincWen2025x,
      author = {Seamus Albion Ferlinc and Joshua Jeishing Wen},
      title = {Ext operators for wreath {M}acdonald polynomials},
      year = {2025},
      eprint = {2508.10772},
      url = {https://arxiv.org/abs/2508.10772},
      journal = {arXiv e-prints}
    }
    

15. [GL19]Camilo González and Luc Lapointe. Multi-Macdonald polynomials. arXiv:1909.09354, 2019.
    .bib

    Copy Download

    @article{GonzalesLapointe2019,
    Author = {Camilo González and Luc Lapointe},
    Title = {Multi-{M}acdonald polynomials},
    Year = {2019},
    Eprint = {1909.09354},
      url = {https://arxiv.org/abs/1909.09354},
    journal = {arXiv e-prints}
    }
    

16. [Goo25]Ben Goodberry. Type A partially-symmetric macdonald polynomials. Algebraic Combinatorics, 7(6):1647–1694, 2025.
    .bib

    Copy Download

    @article{Goodberry2025,
      author = {Ben Goodberry},
      title = {Type {A} partially-symmetric Macdonald polynomials},
      year = {2025},
      journal = {Algebraic Combinatorics},
      volume = {7},
      number = {6},
      pages = {1647--1694},
      doi = {10.5802/alco.388},
      url = {https://doi.org/10.5802/alco.388}
    }
    

17. [Gor26]Vadim Gorin. A law of large numbers for Macdonald coherent measures. arXiv:2609.14217v1, 2026.
    .bib

    Copy Download

    @article{Gorin2026x,
      author = {Vadim Gorin},
      title = {A law of large numbers for {M}acdonald coherent measures},
      year = {2026},
      eprint = {2609.14217v1},
      url = {https://arxiv.org/abs/2609.14217v1},
      journal = {arXiv e-prints}
    }
    

18. [GH26]Alice Guionnet and Jiaoyang Huang. Asymptotics of Symmetric Polynomials: A Dynamical Point of View. Communications in Mathematical Physics, 407(7), 2026.
    .bib

    Copy Download

    @article{GuionnetHuang2026,
      author = {Guionnet, Alice and Huang, Jiaoyang},
      title = {Asymptotics of {S}ymmetric {P}olynomials: {A} {D}ynamical {P}oint of {V}iew},
      year = {2026},
      journal = {Communications in Mathematical Physics},
      volume = {407},
      number = {7},
      publisher = {Springer Science and Business Media LLC},
      doi = {10.1007/s00220-026-05639-9},
      url = {http://dx.doi.org/10.1007/s00220-026-05639-9},
      issn = {1432-0916}
    }
    

19. [Hag10]Jim Haglund. Combinatorics associated to type $A$ nonsymmetric Macdonald polynomials. Oberwolfach report, 2010.
    .bib

    Copy Download

    @misc{Haglund2010,
     author = {Jim Haglund},
     title = {Combinatorics Associated to type $A$ nonsymmetric {M}acdonald polynomials},
     howpublished = {Oberwolfach report},
     url = {https://www2.math.upenn.edu/~jhaglund/conf/MFOtalk.pdf},
     year = {2010}
    }
    

20. [HHL05]James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc., 18(03):735–762, July 2005.
    .bib

    Copy Download

    @article{HaglundHaimanLoehr2005,
      doi = {10.1090/s0894-0347-05-00485-6},
      title = {A combinatorial formula for {M}acdonald polynomials},
      year  = {2005},
      month = jul,
      publisher = {American Mathematical Society ({AMS})},
      volume = {18},
      number = {03},
      pages = {735--762},
      author = {James Haglund and Mark D. Haiman and Nicholas A. Loehr},
      journal = {J. Amer. Math. Soc.}
    }
    

21. [HHL08]James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for nonsymmetric Macdonald polynomials. American Journal of Mathematics, 130(2):359–383, 2008.
    .bib

    Copy Download

    @article{HaglundHaimanLoehr2008,
     ISSN = {00029327, 10806377},
     URL2 = {http://www.jstor.org/stable/40068131},
     doi = {10.1353/ajm.2008.0015},
     author = {James Haglund and Mark D. Haiman and Nicholas A. Loehr},
     journal = {American Journal of Mathematics},
     number = {2},
     pages = {359--383},
     publisher = {The Johns Hopkins University Press},
     title = {A combinatorial formula for nonsymmetric {M}acdonald polynomials},
     volume = {130},
     year = {2008}
    }
    

22. [Hai01]Mark D. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society, 14(04):941–1007, October 2001.
    .bib

    Copy Download

    @article{Haiman2001,
      doi = {10.1090/s0894-0347-01-00373-3},
      year  = {2001},
      month = oct,
      publisher = {American Mathematical Society ({AMS})},
      volume = {14},
      number = {04},
      pages = {941--1007},
      author = {Mark D. Haiman},
      title = {Hilbert schemes, polygraphs and the {M}acdonald positivity conjecture},
      journal = {Journal of the American Mathematical Society}
    }
    

23. [Hai02]Mark Haiman. Combinatorics, symmetric functions and Hilbert schemes. Current developments in mathematics, 2002(1):39–111, 2002.
    .bib

    Copy Download

    @article{Haiman2002hilbertSchemes,
      title={Combinatorics, symmetric functions and {H}ilbert schemes},
      author={Mark Haiman},
      journal={Current developments in mathematics},
      volume={2002},
      number={1},
      pages={39--111},
      year={2002},
      publisher={International Press of Boston}
    }
    

24. [He20]Jimmy He. A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field. International Mathematics Research Notices, 2022(9):6854–6902, 2020.
    .bib

    Copy Download

    @article{He2020,
      author = {He, Jimmy},
      title = {A {C}haracteristic {M}ap for the {S}ymmetric {S}pace of
        {S}ymplectic {F}orms {O}ver a {F}inite {F}ield},
      year = {2020},
      journal = {International Mathematics Research Notices},
      volume = {2022},
      number = {9},
      pages = {6854--6902},
      publisher = {Oxford University Press (OUP)},
      doi = {10.1093/imrn/rnaa309},
      url = {http://dx.doi.org/10.1093/imrn/rnaa309},
      issn = {1687-0247},
      eprint = {1906.05966}
    }
    

25. [JL26]Emma Yu Jin and Xiaowei Lin. Combinatorial formulas for Macdonald polynomials by superizations. arXiv:2602.12672v4, 2026.
    .bib

    Copy Download

    @article{JinLin2026x,
      author = {Emma Yu Jin and Xiaowei Lin},
      title = {Combinatorial formulas for {M}acdonald polynomials by superizations},
      year = {2026},
      eprint = {2602.12672v4},
      url = {https://arxiv.org/abs/2602.12672v4},
      journal = {arXiv e-prints}
    }
    

26. [Kos21]Shinji Koshida. Free field theory and observables of periodic Macdonald processes. Journal of Combinatorial Theory, Series A, 182:105473, 2021.
    .bib

    Copy Download

    @article{Koshida2021,
      author = {Shinji Koshida},
      title = {Free field theory and observables of periodic {M}acdonald
        processes},
      year = {2021},
      journal = {Journal of Combinatorial Theory, Series A},
      volume = {182},
      pages = {105473},
      doi = {10.1016/j.jcta.2021.105473},
      eprint = {2001.04607}
    }
    

27. [Lan09]Robin Langer. Macdonald polynomials and symmetric functions. arXiv:0907.3950, 2009.
    .bib

    Copy Download

    @article{Langer2009,
    Author = {Robin Langer},
    Title = {Macdonald polynomials and symmetric functions},
    Year = {2009},
    Eprint = {0907.3950},
      url = {https://arxiv.org/abs/0907.3950},
    journal = {arXiv e-prints}
    }
    

28. [Lap22]Luc Lapointe. $m$-symmetric functions, non-symmetric macdonald polynomials and positivity conjectures. arXiv:2206.05177, 2022.
    .bib

    Copy Download

    @article{Lapointe2022x,
      author = {Luc Lapointe},
      title = {{$m$}-symmetric functions, non-symmetric Macdonald polynomials and
        positivity conjectures},
      year = {2022},
      eprint = {2206.05177},
      url = {https://arxiv.org/abs/2206.05177},
      journal = {arXiv e-prints}
    }
    

29. [LLM98]Luc Lapointe, Alain Lascoux and Jennifer Morse. Determinantal expressions for Macdonald polynomials. International Mathematics Research Notices, 1998(18):957, 1998.
    .bib

    Copy Download

    @article{LapointeLascouxMorse1998,
      doi = {10.1155/s1073792898000579},
      url2 = {https://doi.org/10.1155/s1073792898000579},
      year  = {1998},
      publisher = {Oxford University Press ({OUP})},
      volume = {1998},
      number = {18},
      pages = {957},
      title={Determinantal expressions for {M}acdonald polynomials},
      author = {Luc Lapointe and Alain Lascoux and Jennifer Morse},
      journal = {International Mathematics Research Notices}
    }
    

30. [LP26]Luc Lapointe and Luis Pena. A new characterization of right keys, and the $m$-symmetric Schur functions at $t=0$. arXiv:2608.13276v1, 2026.
    .bib

    Copy Download

    @article{LapointePena2026Keysx,
      author = {Luc Lapointe and Luis Pena},
      title = {A new characterization of right keys, and the $m$-symmetric
        {S}chur functions at $t=0$},
      year = {2026},
      eprint = {2608.13276v1},
      url = {https://arxiv.org/abs/2608.13276v1},
      journal = {arXiv e-prints}
    }
    

31. [LP26]Luc Lapointe and Luis Pena. A proof of the $m$-Symmetric Macdonald positivity at $t=1$. arXiv:2608.10223v1, 2026.
    .bib

    Copy Download

    @article{LapointePena2026Macdonaldx,
      author = {Luc Lapointe and Luis Pena},
      title = {A proof of the $m$-{S}ymmetric {M}acdonald positivity at $t=1$},
      year = {2026},
      eprint = {2608.10223v1},
      url = {https://arxiv.org/abs/2608.10223v1},
      journal = {arXiv e-prints}
    }
    

32. [Las05]Michel Lassalle. Some conjectures for Macdonald polynomials of type B, C, D. arXiv:math/0503149, 2005.
    .bib

    Copy Download

    @article{Lassalle2005x,
      author = {Michel Lassalle},
      title = {Some conjectures for {M}acdonald polynomials of type {B}, {C}, {D}},
      year = {2005},
      eprint = {math/0503149},
      url = {https://arxiv.org/abs/math/0503149},
      journal = {arXiv e-prints}
    }
    

33. [LNNS25]Cristian Lenart, Satoshi Naito, Fumihiko Nomoto and Daisuke Sagaki. Symmetric and nonsymmetric Macdonald polynomials via a path model with a pseudo-crystal structure. Macdonald Theory and Beyond:25–57, 2025.
    .bib

    Copy Download

    @article{LenartNaitoNomotoSagaki2025,
      author = {Lenart, Cristian and Naito, Satoshi and Nomoto, Fumihiko and
        Sagaki, Daisuke},
      title = {Symmetric and nonsymmetric {M}acdonald polynomials via a path model
        with a pseudo-crystal structure},
      year = {2025},
      journal = {Macdonald Theory and Beyond},
      pages = {25--57},
      publisher = {American Mathematical Society},
      doi = {10.1090/conm/815/16315},
      url = {https://doi.org/10.1090/conm/815/16315},
      eprint = {2210.14464}
    }
    

34. [Mac88]Ian G. Macdonald. A new class of symmetric functions. Séminaire Lotharingien de Combinatoire [electronic only], 20(B20a), 1988.
    .bib

    Copy Download

    @article{Macdonald1987,
    	author = {Ian G.  Macdonald},
    	journal = {S{\'{e}}minaire Lotharingien de Combinatoire [electronic only]},
    	language = {eng},
    	number = {B20a},
    	publisher = {Universit{\"{a}}t Wien, Fakult{\"{a}}t f{\"{u}}r Mathematik},
    	title = {A new class of symmetric functions},
    	url = {http://eudml.org/doc/120965},
    	volume = {20},
    	year = {1988}
    }
    

35. [Mac95]Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
    .bib

    Copy Download

    @book{Macdonald1995,
    	address = {New York},
    	author = {Ian G. Macdonald},
    	edition = {Second},
    	isbn = {0-19-853489-2},
    	mrclass = {05E05 (05-02 20C30 20C33 20K01 33C80 33D80)},
    	note = {With contributions by A. Zelevinsky, Oxford Science Publications},
    	pages = {x+475},
    	publisher = {The Clarendon Press, Oxford University Press},
    	series = {Oxford Mathematical Monographs},
    	title = {Symmetric functions and {H}all polynomials},
    	year = {1995}
    }
    

36. [Man24]Olya Mandelshtam. Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials. arXiv:2401.17223, 2024.
    .bib

    Copy Download

    @article{Mandelshtam2024x,
      author = {Olya Mandelshtam},
      title = {Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric {M}acdonald polynomials},
      year = {2024},
      eprint = {2401.17223},
      url = {https://arxiv.org/abs/2401.17223},
      journal = {arXiv e-prints}
    }
    

37. [MT22]Dario Mathiä and Ulrich Thiel. Wreath Macdonald polynomials at $q=t$ as characters of rational Cherednik algebras. Transactions of the American Mathematical Society, 375(12):8945–8968, 2022.
    .bib

    Copy Download

    @article{MathiaThiel2022,
      author = {Dario Mathi{\"a} and Ulrich Thiel},
      title = {Wreath {M}acdonald polynomials at $q=t$ as characters of rational
        {C}herednik algebras},
      year = {2022},
      journal = {Transactions of the American Mathematical Society},
      volume = {375},
      number = {12},
      pages = {8945--8968},
      doi = {10.1090/tran/8774},
      url = {https://doi.org/10.1090/tran/8774},
      eprint = {2112.10604}
    }
    

38. [MW23]Konstantin Matveev and Yuchen Wei. Factorization of certain Macdonald Littlewood-Richardson coefficients. arXiv:2301.06500, 2023.
    .bib

    Copy Download

    @article{MatveevWei2023x,
      author = {Konstantin Matveev and Yuchen Wei},
      title = {Factorization of certain {M}acdonald
        {L}ittlewood-{R}ichardson coefficients},
      year = {2023},
      eprint = {2301.06500},
      url = {https://arxiv.org/abs/2301.06500},
      journal = {arXiv e-prints}
    }
    

39. [Mic26]Ryan Mickler. On the Nazarov–Sklyanin Lax operator for Macdonald functions. arXiv:2610.12326, 2026.
    .bib

    Copy Download

    @article{Mickler2026Lax,
      author = {Ryan Mickler},
      title = {On the {N}azarov--{S}klyanin {L}ax operator for {M}acdonald functions},
      year = {2026},
      eprint = {2610.12326},
      url = {https://arxiv.org/abs/2610.12326},
      journal = {arXiv e-prints}
    }
    

40. [Oh22]Jaeseong Oh. Macdonald polynomials and cyclic sieving. European Journal of Combinatorics, 101:103465, 2022.
    .bib

    Copy Download

    @article{Oh2022,
      author = {Jaeseong Oh},
      title = {Macdonald polynomials and cyclic sieving},
      year = {2022},
      journal = {European Journal of Combinatorics},
      volume = {101},
      pages = {103465},
      doi = {10.1016/j.ejc.2021.103465},
      eprint = {2102.09982}
    }
    

41. [Ols21]Grigori Olshanski. Macdonald polynomials and extended Gelfand–Tsetlin graph. Selecta Mathematica, 27(3), 2021.
    .bib

    Copy Download

    @article{Olshanski2021,
      author = {Grigori Olshanski},
      title = {Macdonald polynomials and extended {G}elfand--{T}setlin graph},
      year = {2021},
      journal = {Selecta Mathematica},
      volume = {27},
      number = {3},
      doi = {10.1007/s00029-021-00660-3},
      eprint = {2007.06261}
    }
    

42. [OS23]Daniel Orr and Mark Shimozono. Wreath Macdonald polynomials, a survey. arXiv:2308.12166, 2023.
    .bib

    Copy Download

    @article{OrrShimozono2023x,
    Author = {Daniel Orr and Mark Shimozono},
    Title = {Wreath {M}acdonald polynomials, a survey},
    Year = {2023},
    Eprint = {2308.12166},
      url = {https://arxiv.org/abs/2308.12166},
    journal = {arXiv e-prints}
    }
    

43. [OS24]Daniel Orr and Mark Shimozono. Difference operators for wreath Macdonald polynomials. Categorical, Combinatorial and Geometric Representation Theory and Related Topics:483–492, 2024.
    .bib

    Copy Download

    @misc{OrrShimozono2024,
      author = {Orr, Daniel and Shimozono, Mark},
      title = {Difference operators for wreath {M}acdonald polynomials},
      year = {2024},
      journal = {Categorical, Combinatorial and Geometric Representation Theory and Related Topics},
      pages = {483--492},
      publisher = {American Mathematical Society},
      doi = {10.1090/pspum/108/01970},
      url = {http://dx.doi.org/10.1090/pspum/108/01970},
      issn = {2324-707X},
      isbn = {9781470471170},
      eprint = {2110.08808}
    }
    

44. [OSW25]Daniel Orr, Mark Shimozono and Joshua Wen. Wreath Macdonald operators. Forum of Mathematics, Sigma, 13:e116, 2025.
    .bib

    Copy Download

    @article{OrrShimozonoWen2025,
      author = {Orr, Daniel and Shimozono, Mark and Wen, Joshua},
      title = {Wreath {M}acdonald operators},
      year = {2025},
      journal = {Forum of Mathematics, Sigma},
      volume = {13},
      pages = {e116},
      publisher = {Cambridge University Press},
      doi = {10.1017/fms.2025.10061},
      url = {https://doi.org/10.1017/fms.2025.10061},
      eprint = {2211.03851}
    }
    

45. [OW24]Daniel Orr and Milo Bechtloff Weising. Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes. arXiv:2410.13642, 2024.
    .bib

    Copy Download

    @article{OrrWeising2024x,
      author = {Daniel Orr and Milo Bechtloff Weising},
      title = {Stable-limit partially symmetric {M}acdonald functions and parabolic
        flag {H}ilbert schemes},
      year = {2024},
      eprint = {2410.13642},
      url = {https://arxiv.org/abs/2410.13642},
      journal = {arXiv e-prints}
    }
    

46. [RW25]Marino Romero and Joshua Jeishing Wen. Tesler identities for wreath Macdonald polynomials. arXiv:2505.01732, 2025.
    .bib

    Copy Download

    @article{RomeroWen2025x,
      author = {Marino Romero and Joshua Jeishing Wen},
      title = {Tesler identities for wreath {M}acdonald polynomials},
      year = {2025},
      eprint = {2505.01732},
      url = {https://arxiv.org/abs/2505.01732},
      journal = {arXiv e-prints}
    }
    

47. [Sch26]Philip Schlösser. Intermediate Macdonald Polynomials and Their Vector Versions. Symmetry, Integrability and Geometry: Methods and Applications, 2026.
    .bib

    Copy Download

    @article{Schlosser2026Macdonald,
      author = {Schlösser, Philip},
      title = {Intermediate {M}acdonald {P}olynomials and {T}heir {V}ector {V}ersions},
      year = {2026},
      journal = {Symmetry, Integrability and Geometry: Methods and Applications},
      publisher = {SIGMA (Symmetry, Integrability and Geometry: Methods and Application)},
      doi = {10.3842/sigma.2026.057},
      url = {http://dx.doi.org/10.3842/SIGMA.2026.057},
      issn = {1815-0659},
      eprint = {2310.17362}
    }
    

48. [Wen25]Joshua Jeishing Wen. Wreath Macdonald polynomials as eigenstates. Selecta Mathematica, 31(3), 2025.
    .bib

    Copy Download

    @article{Wen2025,
      author = {Wen, Joshua Jeishing},
      title = {Wreath {M}acdonald polynomials as eigenstates},
      year = {2025},
      journal = {Selecta Mathematica},
      volume = {31},
      number = {3},
      publisher = {Springer Science and Business Media LLC},
      doi = {10.1007/s00029-025-01059-0},
      url = {http://dx.doi.org/10.1007/s00029-025-01059-0},
      issn = {1420-9020},
      eprint = {1904.05015}
    }
    

49. [Yoo12]Meesue Yoo. A combinatorial formula for the schur coefficients of the integral form of the Macdonald polynomials in the two column and certain hook cases. Annals of Combinatorics, 16(2):389–410, February 2012.
    .bib

    Copy Download

    @article{Yoo2012,
      doi = {10.1007/s00026-012-0133-x},
      url2 = {https://doi.org/10.1007/s00026-012-0133-x},
      year = {2012},
      month = feb,
      publisher = {Springer Science and Business Media {LLC}},
      volume = {16},
      number = {2},
      pages = {389--410},
      author = {Meesue Yoo},
      title = {A Combinatorial Formula for the Schur Coefficients of the Integral Form of the {M}acdonald Polynomials in the Two Column and Certain Hook Cases},
      journal = {Annals of Combinatorics}
    }
    

50. [YOO15]Meesue YOO. Schur coefficients of the integral form Macdonald polynomials. Tokyo Journal of Mathematics, 38(1), June 2015.
    .bib

    Copy Download

    @article{Yoo2015,
      doi = {10.3836/tjm/1437506242},
      url2 = {https://doi.org/10.3836/tjm/1437506242},
      year = {2015},
      month = jun,
      publisher = {Tokyo Journal of Mathematics},
      volume = {38},
      number = {1},
      author = {Meesue YOO},
      title = {Schur Coefficients of the Integral Form {M}acdonald Polynomials},
      journal = {Tokyo Journal of Mathematics}
    }