There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here "symmetric" means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.
Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to superpolynomials. The theory significantly differs from the supersymmetric Jack polynomials introduced and studied in several papers by Desrosiers, Mathieu and Lapointe (Nucl. Phys. B606, 2001). The vector-valued Jack polynomials arise in standard modules of the rational Cherednik algebra and were originated by Griffeth (T.A.M.S. 362, 2010) for the family G(n,p,N) of com…
Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase partitions and are associated to special parameter values $(q,t)$. For $N$ variables, there are singular polynomials for any pair of positive integers $m$ and $n$, with $2\leq n\leq N$, and parameters values $(q,t)$ satisfying $q^{a}t^{b}=1$ exactly when $a=rm$ and $b=rn$, for some integer $r$. The coefficients of nonsymmetric Macdonald polynomials with respect to the basis of monomials $\big\{ x^α\big\}$ …
The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular poly…
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 (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.
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.
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.
The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to $\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$ and $A_2^{(2)\dagger}$. We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules.
We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at $t=0$ in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crysta…
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…
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…
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 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…
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 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…
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…
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…
Recent studies in several interrelated areas -- from combinatorics and representation theory in mathematics to quantum field theory and topological string theory in physics -- have independently revealed that many classical objects in these fields admit a relatively novel one-parameter deformation. This deformation, known in different contexts under the names of Omega-background, refinement, or beta-deformation, has a number of interesting mathematical implications. In particular, in Chern-Simons theory beta-deformation transforms the classical HOMFLY invariants into Dunfield-Gukov-Rasmussen s…
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 study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules.
When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q-analogues of this fact were conjectured in math.QA/0112035; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies…
Koornwinder polynomials are a 6-parameter BC_n-symmetric family of Laurent polynomials indexed by partitions, from which Macdonald polynomials can be recovered in suitable limits of the parameters. As in the Macdonald polynomial case, standard constructions via difference operators do not allow one to directly control these polynomials at q=0. In the first part of this paper, we provide an explicit construction for these polynomials in this limit, using the defining properties of Koornwinder polynomials. Our formula is a first step in developing the analogy between Hall-Littlewood polynomials …
This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and the symmetrization of the Macdonald polynomials. The main tool of the study is the Yang-Baxter graph. We show that these Macdonald polynomials can be easily computed following this graph. We give als…
We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of the representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partiti…
We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.
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.
In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group $\mathcal{S}_{N}$. The polynomials are parametrized by $\left( q,t\right) $ and are simultaneous eigenfunctions of a commuting set of Cherednik operators, which were studied by Baker and Forrester (IMRN 1997). In the Dunkl-Luque paper there is a construction of a pairing between $\left( q^{-1},t^{-1}\right) $ polynomials and $\left( q,t\right) $ polynomials, and for which the Macdon…
For each partition $τ$ of $N$ there are irreducible modules of the symmetric groups $\mathcal{S}_{N}$ or the corresponding Hecke algebra $\mathcal{H}_{N}\left( t\right) $ whose bases consist of reverse standard Young tableaux of shape $τ$. There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules, respectively.The Jack polynomials are a special case of those constructed by Griffeth for the infinite family $G\left( n,p,N\right) $ of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For both the group $\…
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}.
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…
Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra
HSE University（高等经济大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Yanqi Lake Beijing Institute of Mathematical Sciences And Applications（北京雁栖湖应用数学研究院）；Virginia Tech（弗吉尼亚理工大学）；Max Planck Institut fur Mathematik（马克斯·普朗克数学研究所）
We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the so-called splice diagrams. The specialization t=q of our superpolynomials conjecturally results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the correspond…
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…
We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between orient…
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check t…
