Andrei Okounkov
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.
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We prove a binomial formula for Macdonald polynomials and consider applications of it.
A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl_3 Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl_3 series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl_3 Selberg integral for Jack p…
Inhomogeneous analogues of symmetric and nonsymmetric Macdonald polynomials were introduced by F. Knop and the author. In the symmetric case A. Okounkov has recently proved a beautiful expansion formula which can be viewed as a multivariable generalization of the q-binomial theorem. It is becoming clear that many interesting results in the symmetric setting can also be established in the nonsymmetric setting. In this paper, we prove the nonsymmetric analogue the binomial formula, and also discuss the limiting case of Jack polynomials.
We consider 3-parametric polynomials which replace the A-series interpolation Macdonald polynomials in the BC case. For these polynomials, we prove: an integral representation, a combinatorial formula, Pieri-type rules, Cauchy identity, and we also show that they do not satisfy any rational q-difference equation. We also prove a binomial formula for the 6-parametric Koornwinder polynomials.
Raising operators of row type are constructed by means of an interpolation method. These are a dual version of the raising operators of column type by A.N.Kirillov and M.Noumi. An extension of the q-binomial coefficients is introduced in relation to the raising operators.
We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.
The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial $P_κ(z)$ are known explicitly. These formulas generalise the known $r=1$ case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials $E_η(z)$. In this paper we extend beyond the case $r=1$ for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{η}{ν}_{q,t}$ associated with the n…
In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the cases r=1 and r=n-1. We do this by first deducing the the decomposition for the product of any nonsymmetric Macdonald polynomial with a linear function in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of the product of the first elementary function with a nonsymmetric Macdonald polynomial we evaluate the genera…
Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities
本文研究插值多项式及其关联系数，证明了二项系数的正性和单调性，以及Littlewood-Richardson系数的部分正性，并提出一个新的对称函数不等式，建立Jack展开正性与分拆包含顺序之间的对偶关系。
This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules $ {\bf M}_μ$ and corresponding elements of the Macdonald basis. We recall that ${\bf M}_μ$ is defined for a partition $μ\part n$, as the linear span of derivatives of a certain bihomogeneous polynomial $Δ_ μ(x,y)$ in the variables $x_1,x_2,..., x_n, y_1,y_2,..., y_n$. It has been conjectured by Garsia and Haiman that ${\bf M}_μ$ has $n!$ dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial ${\widetilde{H}}_μ(x;q,t)=\sum_{λ\part n} S_λ(X) {\widet…
A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.
A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials
引入单行Macdonald多项式的多参数形变算子，提出统一的多参数Murnaghan-Nakayama规则，推导了多种多项式的迭代公式并简化了相关代数的证明。
We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence $A^1, A^2, \ldots$ of commuting operators that are diagonalized by them. Such a sequence of operators arises in the projective limit of finite families of commuting q-difference operators studied by Okounkov, Knop and Sahi. The main theorem is an explicit formula for the operators $A^k$. Our formula involves the family of Hall-Littlewood functions and a new family of inhomogeneous Hall-Littlewood functions, for which we give an explicit construction and identify as a …
Wreath Macdonald polynomials, a survey
本文概述了Wen关于Wreath Macdonald多项式的研究成果，探讨了其几何基础和反射 functor 的作用，并提出了新的猜想。
We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of $K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the integer level representation of the Ding-Iohara quantum algebra naturally produces the currents of the deformed $\mathcal{W}$ algebra. Then w…
We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of…
Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by …
Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials
本文研究了类型$\mathfrak{gl}_{1|1}$的位移量子扭代数，揭示了超Macdonald多项式与量子扭代数超荷作用之间的关系，并通过微分算子推导出Pieri规则，进一步推导出超对称哈密顿量，恢复了文献中的已有结果。
We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link $L_{8n8}$, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expres…
Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^{(1)}_n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n.
Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions
本文利用量子环面 $\mathfrak{gl}_1$ 代数关系研究 $q,t$-矩阵模型的递归，统一证明并推广 Gaussian、Laguerre、Jacobi 等模型的 Macdonald 多项式超可积性公式。
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