We examine the non-symmetric Macdonald polynomials $E_λ(x;q,t)$ at $q=1$, as well as the more general permuted-basement Macdonald polynomials. When $q=1$, we show that $E_λ(x;1,t)$ is symmetric and independent of $t$ whenever $λ$ is a partition. Furthermore, we show that for general $λ$, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of $t$, while the non-symmetric part only depends on the relative order of the entries in $λ$.
We also examine the case $q=0$, which give rise to so called permuted-basement $t$-atoms. We prove expansion-properties of these, 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 a product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis, and thus interpolates between two results by Haglund, Luoto, Mason and van Willigenburg.
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 the first author.
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 prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_λ$, that was conjectured by S.V. Kerov in 1992.
We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to $q,t$-analogs of the dimensions of the irreduc…
We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials bases and behave well under the Demazure-Lusztig operators. The symmetric Macdonald polynomials $J_λ$ are expressed as a sum of general-basement Macdonald polynomials via an explicit formula. By letting $q=0$, we obtain $t$-deformations of key polynomials and Demazure atoms and we show that the Hall--Littlewood polynomials expand positively into these. This generalizes a result by Haglund, Luoto, Mason…
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 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 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.
In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta (2008), but our method of derivation is to use Cherednik's interwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip (2010). Specializations of the Monk formula's at $q=0$ and/or $t=0$ provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.
This paper is a supplement to Guo-Ram arXiv:2104.02942, containing examples, remarks and additional material that could be useful to researchers working with Type $GL_n$ Macdonald polynomials. In the course of our comparison of the alcove walk formula and the nonattacking fillings formulas for type $GL_n$ Macdonald polynomials we did many examples and significant analysis of the literature. In the preparation of Guo-Ram arXiv:2104.02942 it seemed sensible to produce a document with focus and this material was removed. This is paper resurrects and organizes that material, in hopes that others m…
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.
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…
The paper compares (and reproves) the alcove walk and the nonattacking fillings formulas for type $GL_n$ Macdonald polynomials which were given in Haglund-Haiman-Loehr arXiv:math.CO/0601693, Alexandersson arXiv:1602.05153 and Ram-Yip arXiv:0803.1146. The "compression" relating the two formulas in this paper is the same as that of Lenart arXiv:0804.4716. We have reformulated it so that it holds without conditions and so that the proofs of the alcove walk formula and the nonattacking fillings formula are parallel. This reformulation highlights the role of the double affine Hecke algebra and Cher…
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.
We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a generalization of Cherednik's non-symmetric Macdonald polynomials of type $GL_n$. We utilize links between $^+$stable-limit double affine Hecke algebra theory of Ion and Wu and the double Dyck path algebra of Carlsson and Mellit that arose in their proof of the Shuffle Conjecture. As a consequence, the spectral theory for the limit Cherednik operators is unde…
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 prove a strong factorization property of interpolation Macdonald polynomials when $q$ tends to $1$. As a consequence, we show that Macdonald polynomials have a strong factorization property when $q$ tends to $1$, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate $q,t$-Kostka numbers and we show that they are polynomials in $q,t$ with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ with nonnegative integer c…
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.
We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.
Macdonald characters from a new formula for Macdonald polynomials
Université de Lorraine（洛林大学）；CNRS, IECL（法国国家科学研究中心，IECL）；Université de Paris（巴黎大学）；CNRS, IRIF（法国国家科学研究中心，IRIF）；Harvard University（哈佛大学）；Università di Pisa（比萨大学）
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 demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.
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.
We define a family of symmetric polynomials $G_{ν,λ}(z_1,\cdots, z_{n+1},q)$ indexed by a pair of dominant integral weights. The polynomial $G_{ν,0}(z,q)$ is the specialized Macdonald polynomial and we prove that $G_{0,λ}(z,q)$ is the graded character of a level two Demazure module associated to the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$. Under suitable conditions on $(ν,λ)$ (which includes the case when $ν=0$ or $λ=0$) we prove that $G_{ν,λ}(z,q)$ is Schur positive and give explicit formulae for them in terms of Macdonald polynomials.
The quasisymmetic Macdonald polynomials $G_γ(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_λ(X; q, t)$ with the property that $G_γ(X; 0, 0)$ equals $QS_γ(X)$, the quasisymmetric Schur polynomial of [9]. We derive an expansion for $G_γ(X; q, t)$ in the fundamental basis of quasisymmetric functions.
