Nonsymmetric Macdonald Polynomials Research Articles

Specializing Koornwinder polynomials to Macdonald polynomials of type B, C, D and BC

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type \((C_n^\vee ,C_n)\) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials.

Connections between vector-valued and highest weight Jack and Macdonald polynomials

We analyze conditions under which a projection from the vector-valued Jack orMacdonald polynomials to scalar polynomials has useful properties, specially 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 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 partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.

A combinatorial formula for Sahi, Stokman, and Venkateswaran's generalization of Macdonald polynomials

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 ⌊n/2⌋ “metaplectic parameters”), such that the n=1 case recovers the nonsymmetric Macdonald polynomials and the q→0,∞ limits with a choice of metaplectic parameters (as Gauss sums) yield metaplectic Iwahori Whittaker functions. In this paper, we study these new polynomials, which we call SSV polynomials, in the case of GLr. We apply a result of Ram and Yip in order to give a combinatorial formula for the SSV polynomials and their symmetrized variants in terms of alcove walks. The result is used to precisely characterize the support of each SSV polynomial, showing that SSV polynomials have fewer terms than the corresponding Macdonald polynomials. Finally, we calculate the q→0 and q→∞ limits of the SSV polynomials and observe that our combinatorial formula can be written in terms of alcove walks with only positive and negative folds respectively. We conclude by giving the correspondence between SSV polynomials and metaplectic Iwahori Whittaker functions, part of which is due to Sahi, Stokman, and Venkateswaran.

The Classification of All Singular Nonsymmetric Macdonald Polynomials

The affine Hecke algebra of type A has two parameters q,t 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 polynomials. The possible parameter values are of the form qm=t−n with 2≤n≤N. For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is dn−1,n−1,…,n−1,r for some d≥1. In the present paper, it is shown that there are no other singular polynomials.

Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

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 combinatorially in type A that every affine Demazure module admits a finite Demazure flag.

Nonsymmetric Macdonald Superpolynomials

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.

Demazure crystals for specialized nonsymmetric Macdonald polynomials

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 crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall–Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

Cyclic sieving, skew Macdonald polynomials and Schur positivity

When λ is a partition, the specialized non-symmetric Macdonald polynomial E λ (x;q;0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by E λ (x;q;0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades.

Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules

We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at t=∞ 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 an application, the coefficients of the nonsymmetric Rogers-Ramanujan series provide formulas for the multiplicities of the expansions of tensor products of level-one Kac-Moody representations in terms of Demazure slices.

LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(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 $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules $$ {V}_z^{-}\left(\uplambda \right) $$ for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.

Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

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 $\big\{ x^{\alpha}\big\}$ are rational functions of $q$ and $t$. In this paper, we present the construction of subspaces of singular nonsymmetric Macdonald polynomials specialized to particular values of $(q,t)$. The key part of this construction is to show the coefficients have no poles at the special values of $(q,t)$. Moreover, this subspace of singular Macdonald polynomials for the special values of the parameters is an irreducible module for the Hecke algebra of type $A_{N-1}$.

Some Remarks on Non-Symmetric Interpolation Macdonald Polynomials

Abstract 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 to binomial formulas involving non-symmetric interpolation Macdonald polynomials.

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for V_n-valued meromorphic functions on a complex n-torus, with V_n a module over the mathrm{GL}_n-type extended affine Hecke algebra {mathcal {H}}_n. The family ({mathcal {H}}_n)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {mathcal {H}}_{n+1}, in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers (f^{(n)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 0}. The compatibility is encoded by the so-called braid recursion relations: f^{(n+1)}(z_1,ldots ,z_{n},0) is required to coincide up to a quasi-constant factor with the push-forward of f^{(n)}(z_1,ldots ,z_{n}) by an intertwiner mu _{n}{:},V_{n}rightarrow V_{n+1} of {mathcal {H}}_{n}-modules, where V_{n+1} is considered as an {mathcal {H}}_{n}-module through the tower structure on ({mathcal {H}}_n)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 0} of solutions. The solutions f^{(n)} are constructed from specialized dual non-symmetric Macdonald polynomials with specialized parameters using the Cherednik–Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f^{(n)} coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n.

Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths

Let λ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let QLS(λ) denote the set of quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t=0 and t=∞ of the nonsymmetric Macdonald polynomial Ewλ(q,t) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ:QLS(λ+μ)→QLS(λ)⊗QLS(μ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

Abstract We study the non-symmetric 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 non-symmetric 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 for simply-laced Dynkin types except for type E 8 {\mathrm{E}_{8}} .

Properties of Non-symmetric Macdonald Polynomials at q=1 and q=0

We examine the non-symmetric Macdonald polynomials mathrm {E}_lambda at q=1, as well as the more general permuted-basement Macdonald polynomials. When q=1, we show that mathrm {E}_lambda (mathbf {x};1,t) is symmetric and independent of t whenever lambda is a partition. Furthermore, we show that, in general lambda , this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on mathbf {x}, t, and the relative order of the entries in lambda . We also examine the case q=0, which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.

Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials

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 the Kostka–Foulkes polynomials.

Generalized Weyl modules and nonsymmetric q-Whittaker functions

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.

Demazure character formula for semi-infinite flag varieties

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of global sections of a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin-Makedonskyi's generalized Weyl modules, and the $t = \infty$ specialization of non-symmetric Macdonald polynomials.

REPRESENTATION-THEORETIC INTERPRETATION OF CHEREDNIK-ORR’S RECURSION FORMULA FOR THE SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT T = ∞

We give a representation-theoretic (or rather, crystal-theoretic) proof of Cherednik-Orr's recursion formula of Demazure type for the specialization at t = ∞ of the nonsymmetric Macdonald polynomials Ewλ(q, t), w ∈ W, where λ is a dominant integral weight and W is a finite Weyl group.