J. Haglund, M. Haiman, N. Loehr
We give a combinatorial formula for the non-symmetric Macdonald polynomials E_μ(x;q,t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J_μ(x;q,t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.
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We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t) which had been conjectured by the first author. Corollaries to our main theorem include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients K_{lambda,mu}(q,t) in the case that mu is a partition with parts…
We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on $d$ additional parameters and specialize to all Macdonald polynomials of degree $d$. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the $R$-matrix construction of quantum immanants.
A combinatorial formula for Interpolation Macdonald polynomials
本文给出了插值 Macdonald 多项式的带符号多线队列组合公式，推广了 Macdonald 多项式的多线队列公式。
We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomial…
It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald-Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced, the combinatorial formulas for them are given.
Combinatorial proof of a permuted basement Macdonald polynomial identity
研究置换基 MacDonald 多项式恒等式，通过将 Concha - Lapointe 恒等式细化到特定子族，给出组合证明，并揭示其与 Kazhdan - Lusztig 对合的等价关系。
In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynoimals of type GL(n). At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent-Littelmann).
Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_μ(\mathbf{x};q,t) = \widetilde{H}_{μ^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $μ$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $μ$. We also provide a proof for the full relation in the case when $μ$ is a hook shape, and for all shapes at the speci…
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, whi…
We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis.
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced) superpolynomial of these links.
A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdon…
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials…
We give an explicit Pieri formula for Macdonald polynomials attached to the root system C_n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.
Some combinatorial interpretations of the Macdonald identities for affine root systems
本文利用整数向量与整数划分的联系，通过辛Schur函数和特殊正交Schur函数，给出了七类无限族仿射根系Macdonald恒等式的组合解释，并推导了相关的q-Nekrasov--Okounkov公式。
We give a concise direct proof of the orthogonality of interpolation Macdonald polynomials with respect to the Fourier pairing and briefly discuss some immediate applications of this orthogonality, such as the symmetry of the Fourier pairing and the binomial formula.
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed. In this paper, we present combinatorial formulas for the well poised Macdonald functions, the multiple binomial coefficients, the multiple bracket function, and the multiple Catalan and Lah numbers.
We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula.
We consider products of two Macdonald polynomials of type A, indexed by dominant weights which are respectively a multiple of the first fundamental weight and a weight having zero component on the k-th fundamental weight. We give the explicit decomposition of any Macdonald polynomial of type A in terms of this basis.
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type extensions. Among these the $BC$-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two…
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.
A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle and Schlosser's formula for partitions with length three.
Combinatorial formulas for Macdonald polynomials by superizations
本文通过超填充上的新统计量，推导了对称和非对称Macdonald多项式的最简组合公式，给出了单项式展开系数的显式(q,t)-公式，并推广了Corteel等人的结果。
We derive combinatorial formulae for the modified Macdonald polynomial $H_λ(x;q,t)$ using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to the quantum group of $U_{q}(\wide{sl}_{n+1})$.
We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n, B_n and C_n. This description is in terms of Young tableaux and arises from identifying one-skeleton galleries that appear in the Gaussent-Littelmann formula with Young tableaux. Furthermore, we show by using these tools that the Gaussent-Littelmann formula and the well-known Macdonald formula for Hall-Littlewood polynomials for type A_n are the same.
Sahi, Stokman, and Venkateswaran have constructed, for each positive integer $n$, a family of Laurent polynomials depending on parameters $q$ and $k$ (in addition to $\lfloor n/2\rfloor$ "metaplectic parameters"), such that the $n=1$ case recovers the nonsymmetric Macdonald polynomials and the $q\rightarrow\infty$ limit yields metaplectic Iwahori-Whittaker functions with arbitrary Gauss sum parameters. In this paper, we study these new polynomials, which we call SSV polynomials, in the case of $GL_r$. We apply a result of Ram and Yip in order to give a combinatorial formula for the SSV polynom…
We study the specialization of the type A nonsymmetric Macdonald polynomials at $t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of …
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These "quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These quasisymmetric Macdonald polynomials refine the (symmetric) Macdonald polynomials and specialize at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg. This is an extended abstract.
In a seminal paper Richard Stanley derived Pieri rules for the Jack symmetric function basis. These rules were extended by Macdonald to his now famous symmetric function basis. The original form of these rules had a forbidding complexity that made them difficult to use in explicit calculations. In the early 90's it was discovered that, due to massive cancellations, the dual rule, which expresses skewing by $e_1$ the modified Macdonald polynomial ${\tilde H}_μ[X;q,t]$, can be given a very simple combinatorial form in terms of corner weights of the Ferrers' diagram of $μ$. A similar formula was …
We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a corollary to the result that generalized dual equivalence classes of permutations are unions of standard dual equivalence classes of permutations for certain cases, establishing an earlier conjecture of the author.
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this Note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n =1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.
Recently James Martin introduced multiline queues, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a model of particles hopping on a one-dimensional lattice, which was introduced around 1970, and has been extensively studied in statistical mechanics, probability, and combinatorics. In this article we give an independent proof of Martin's result, and we show that by introducing additional statistics on multiline queues, we can use them to give a new combinatorial formu…
Macdonald characters from a new formula for Macdonald polynomials
本文引入了一个新的算子Γ，用于获得Macdonald多项式生成公式，连接了Macdonald算子理论与移位Macdonald多项式，并提出关于广义特征的正性猜想。
Five-Term Relations for wreath Macdonald polynomials and tableau formulas for Pieri coefficients
本文提出了wreath Macdonald多项式理论中算子的新恒等式，特别是类似于Garsia和Mellit的五项关系，并由此推导出能快速计算单项式展开的wreath Macdonald Pieri系数表公式。
Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas
通过整数向量与双无限词表示的整数分拆之间的关联，给出仿射根系Macdonald恒等式的组合解释，并推导出各类型的q-Nekrasov--Okounkov公式。
Generalizations of the flag variety tied to the Macdonald-theoretic delta operators
该著作章节系统梳理了与 Macdonald 理论 delta 算子相关的旗簇推广，涵盖对称函数恒等式及其表示论与几何表现形式。
We give a direct combinatorial proof of the $q,t$-symmetry relation $\tilde H_μ(X;q,t)=\tilde H_{μ'}(X;t,q)$ in the Macdonald polynomials $\tilde H_μ$ at the specialization $q=1$. The bijection demonstrates that the Macdonald inv statistic on the permutations of any given row of a Young diagram filling is Mahonian. Moreover, our bijection gives rise a family of new bijections on words that preserves the classical Mahonian inv statistic.
We present an explicit branching formula for the six-parameter Macdonald-Koornwinder polynomials with hyperoctahedral symmetry.
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Hall-Littlewood, Jack and Macdonald polynomials. We also give a simple proof of the Jing-Jözefiak formula for two-row Macdonald functions.
Set-valued tableaux formulas play an important role in Schubert calculus. Using the box greedy reduced word for the construction of the Macdonald polynomials, we convert the alcove walk formula for Macdonald polynomials to a set-valued tableaux formula for Macdonald polynomials. Our hope is that providing set-valued tableaux formulas for Macdonald polynomials will help to strengthen the analogies and possible connections between the calculus of Macdonald polynomials and Schubert calculus.
The aim of this note is to give some factorization formulas for different versions of the Macdonald polynomials when the parameter t is specialized at roots of unity, generalizing those existing for Hall-Littlewood functions.
On the Nazarov-Sklyanin Lax operator for Macdonald functions
该研究扩展了杰克情形下的谱理论，将NS算子推广为带辅助变量多项式系数对称函数的Lax算子，得到麦克唐纳小伍德-理查森系数的双参数盒和恒等式及相关线性关系。
We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, S_λ[X(t-1)/(q-1)], can be reduced to addition in λ-rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, S_λ[X], and the monomial basis, m_λ[X].
This paper gives bijective proofs of some novel coinversion identities first discovered by Ayyer, Mandelshtam, and Martin (arxiv:2011.06117) as part of their proof of a new combinatorial formula for the modified Macdonald polynomials $\tilde{H}_μ$. Those authors used intricate algebraic manipulations of $q$-binomial coefficients to prove these identities, which imply the existence of certain bijections needed in their proof that their formula satisfies the axioms characterizing $\tilde{H}_μ$. They posed the open problem of constructing such bijections explicitly. We resolve that problem here.
Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study.
A proof of the $m$-Symmetric Macdonald positivity at $t=1$
本文在t=1的情况下，通过对Kostka系数K_{ΩΛ}(q,1)给出基于m-分拆Ω对应图标准填充的组合解释，证明了原始Macdonald正性猜想向m-对称领域的扩展。
Tesler identities for wreath Macdonald polynomials
本研究利用量子环面代数与shuffle代数生成的圈积Macdonald多项式本征算子，给出将其映射到配分特征δ函数的算子显式公式，证明了互反性相关新结果，还开展了圈积插值Macdonald多项式研究并推导相关系数公式与双谱问题解。
Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $μ= (μ_1,\dots,μ_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_μ$. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such pat…
We present explicit Pieri formulas for Macdonald's spherical functions (or generalized Hall-Littlewood polynomials associated with root systems) and their $q$-deformation the Macdonald polynomials. For the root systems of type $A$, our Pieri formulas recover the well-known Pieri formulas for the Hall-Littlewood and Macdonald symmetric functions due to Morris and Macdonald as special cases.
This is a paper about $c$-functions and Macdonald polynomials. There are $c$-function formulas for $E$-expansions of $P_λ$ and $A_{λ+ρ}$, principal specializations of $P_λ$ and $E_μ$, for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from Macdonald (2003). In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials.
We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. We then discuss a family of polynomials called Demazure atoms. We review the known characterizations of these polynomials and then present two new characterizations. Finally, we consider a family of polynomials called permuted basement nonsymmetric Macdonald polynomials which are obtained by permuting the basement of the combinatorial formula of Haglund, Haiman, and Loehr for nonsymmetric Macdonal…
We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ${}_{12}W_{11}$ series. We remark that the correlation functions of the deformed $\mathcal{W}$ algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.
A raising operator formula for Macdonald polynomials
本文利用LLT多项式上的 $\nabla$ 公式和Haglund-Haiman-Loehr公式，得到修正Macdonald多项式的显式升算子公式，并推广提出 $1,n$-Macdonald多项式及其Schur系数正性猜想。
In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a parameter $t$, and are a specialization of Macdonald polynomials, and further specialize to Schur polynomials upon setting $t=0$. There are three types of formulas for these polynomials. The first formula is in terms of so-called alcove walks, works in arbitrary Lie type, and is derived from the Ram-Yip formula for Macdonald polynomials. The second one is in te…
Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. …
Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials
通过双射证明，建立了置换基底非对称Macdonald多项式在基底交换和形状交换下的四种恒等式，推广了Alexandersson (2019)的结果，并给出了展开规则。
Non-attacking skyline fillings were used by Haglund, Haiman and Loehr to establish a combinatorial formula for nonsymmetric Macdonald polynomials. Semistandard skyline fillings are non-attacking skyline fillings with both major index and coinversion number equal to zero, which serve as a combinatorial model for key polynomials. In this paper, we construct an involution on semistandard skyline fillings. This involution can be viewed as a vast generalization of the classical Bender--Knuth involution. As an application, we obtain that semistandard skyline fillings are compatible with the Demazure…
We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to $t=0$ are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges recovers the finite Demazure crystals constructed earlier by the authors. We also realize the filtration on highest weight modules by Demazure modules by defining explicit embedding operators which, at the level of characters, parallels the recursion operators of Knop and Sahi for specialized nonsymmetric Macdonald polynomials. Thus we prove combinatorial…
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