Bogdan Ion
We establish a connection between (degenerate) nonsymmetric Macdonald polynomials and standard bases and dual standard bases of maximal parabolic modules of affine Hecke algebras. Along the way we prove a (weak) polynomiality result for coefficients of symmetric and nonsymmetric Macdonald polynomials.
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We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p--adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p--adic type.
We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra.
This work records the details of the Ram-Yip formula for nonsymmetric Macdonald-Koornwinder polynomials for the double affine Hecke algebras of not-necessarily-reduced affine root systems. It is shown that the t=0 equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type the proof requires the Ram-Yip formula for the nonsymmetric Koornwinde…
The Cherednik-Orr conjecture expresses the $t\to\infty$ limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.
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…
We establish the equality of the specialization $E_{wλ}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{wλ}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(λ)$ of a certain Demazure-type submodule $U_{w}^{+}(λ)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $λ$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_λ(x;q,0)$ of the symmetric Macdonald polynomial $P_λ(x;q,t)$ at $t=0$ and the grad…
In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ is a dominant integral weight, and $w_{\circ}$ denotes the longest element in the finite Weyl group $W$. Also, for each $x \in W$, we obtain a combinatorial formula for the specialization $E_{x λ} (q, \infty)$ at $t=\infty$ of th…
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…
We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $\hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q…
In this paper we present a Mathematica notebook for computing nonsymmetric and interpolation Macdonald polynomials. We present the new recursive generation algorithm employed within the notebook and the theory required for its development. We detail the contents of the notebook and conclude with a couple of applications of the notebook.
Let $λ\in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_λ$, where $W_λ$ is the stabilizer of $λ$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(λ)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mat…
In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in standard monomial theory), which we call pseudo-quantum Lakshmibai-Seshadri (LS) paths. This model carries less information than the alcove walks in the Ram-Yip formula, and it is therefore more efficient. Furthermore, we construct a connected pseudo-crystal structure on the pseudo-quantum LS paths, which is expected to lead to a simple Littlewood-Richardso…
We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.
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.
Nonsymmetric interpolation Laurent polynomials in $n$ variables are introduced, with the interpolation points depending on $q$ and on a $n$-tuple of parameters $τ=(τ_1,\ldots,τ_n)$. When $τ_i=st^{n-i}$ Okounkov's $3$-parameter $BC_n$-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type $C_n$ Hecke algebra action. In the appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two.
Motivated by the work of Koornwinder, Macdonald, Cherednik, Noumi, and van Diejen we define a 6-parameter double affine Hecke algebra and establish its basic structural properties, including the existence of an involution. We relate the algebra to (symmetric and nonsymmetric) Koornwinder polynomials via the method of intertwiners and, as a consequence, obtain a proof of Macdonald's duality conjecture for Koornwinder polynomials. Combined with earlier work of van Diejen this completely settles all the outstanding conjectures of Macdonald and Koornwinder about these polynomials.
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.
We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric $q$-Whittaker functions. In particular, we show that the series part of the nonsymmetric $q$-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.
We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at $t=\infty$ divided by their norms. The latter are identified with the graded characters of Demazure slices, some canonical quotients of thick (upper) level-one Demazure modules, directly related to recent theory of generalized (nonsymmetric) global Weyl modules. The symmetric Rogers-Ramanujan-type series considered by Cherednik-Feigin were expected to have some interpretation of this kind; the nonsymmetric setting appeared necessary to achieve this. As …
Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr-Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.
We study a Laurent polynomial representation $V$ of the double affine Hecke algebra of type $GL_n$ for specialized parameters $t^{k+1}q^{r-1}=1$. We define a series of subrepresentations of $V$ by using a vanishing condition. For some cases, we give an explicit basis of the subrepresentation in terms of nonsymmetric Macdonald polynomials. These results are nonsymmetric versions of \cite{FJMM} and \cite{KMSV}.
The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack polynomial theory we proceed to derive an expansion formula and a related normalisation. Eigenoperator methods are used to relate the symmetric and antisymmetric Macdonald polynomials, and we discuss how these methods can be extended to special classes of the prescribed symmetry polynomials in terms of their symmetric counterpart. We compute the explicit form of …
Categorification of DAHA and Macdonald polynomials
本文提出与仿射李代数相关的Double Affine Hecke Algebra及其多项式表示和Macdonald多项式的范畴化，通过导出范畴中的函子复合范畴化Demazure算子，并构造模复形使其欧拉示性数等于Macdonald多项式。
Parahoric Lie algebras and parasymmetric Macdonald polynomials
本文旨在对特化的寄生对称 Macdonald 多项式进行范畴化，通过引入仿射 Kac-Moody 李代数的 parahoric 子代数的循环模，证明相关表示范畴为分次最高权范畴，并导出了描述 parahoric 与抛物李群上双模函数的 Peter-Weyl 定理类比。
Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limit t=infinity and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebra for any root system…
A Geometric Realization of Partially-Symmetric Macdonald Polynomials
本文提出猜想将部分对称Macdonald多项式与抛物线旗Hilbert概形联系，扩展了Haiman的几何实现，并证明其与代数$\mathbb{A}_{t,q}$元素作用相容作为证据。
Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes
本文显式计算了抛物线旗Hilbert概形正规化不动点类在同构映射下的像，证明其与修正的部分对称Macdonald多项式一致，从而证实了先前猜想，并给出了对合作用的显式公式。
Quasi-polynomial representations of double affine Hecke algebras
本文引入双仿射海克代数的拟多项式表示，推广 Cherednik 算子与 Macdonald 多项式，并联系 metaplectic 覆盖的表示论，恢复 Iwahori-Whittaker 函数。
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。
We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction …
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