Demazure crystals for specialized nonsymmetric Macdonald polynomials☆
Introduction
Macdonald [33] defined symmetric functions with two parameters indexed by partitions as the unique symmetric function basis satisfying certain triangularity (with respect to monomials in infinitely many variables ) and orthogonality (with respect to a generalized Hall inner product) conditions. The Macdonald symmetric functions give a simultaneous generalization of Hall–Littlewood symmetric functions and Jack symmetric functions .
The coefficients of when written as a sum of monomials are rational functions in the parameters q and t. Macdonald conjectured that the Kostka–Macdonald coefficients defined by expanding the integral form , a scalar multiple of the original , into the plethystic Schur basis, are polynomials in q and t with nonnegative integer coefficients. Here the square brackets denote plethystic substitution. In short, means applied as a Λ-ring operator to the expression A, where Λ is the ring of symmetric functions. For details, see [35] (I.8).
Inspired by work of Garsia and Procesi [14] on Hall–Littlewood symmetric functions, Garsia and Haiman [15] constructed a bi-graded module for the symmetric group and conjectured that the Frobenius character is Thus, the Kostka–Macdonald coefficients give the Schur function expansion of . This conjecture gives a representation theoretic interpretation for the Kostka-Macdonald polynomials as the graded coefficients of the irreducible decomposition of these modules. Haiman [19] resolved both conjectures by analyzing the isospectral Hilbert scheme of points in a plane, ultimately showing that it is Cohen-Macaulay.
The nonsymmetric Macdonald polynomials are indexed by weak compositions and form a basis for the full polynomial ring. They generalize Macdonald polynomials in the sense that where denotes the composition obtained by prepending m 0's to a. The shift to the full polynomial ring begun by Opdam [38], continued by Macdonald [36], and generalized by Cherednik [10] broadened the existing theory in the hopes that the additional structure of the polynomial ring would shed more light on these important functions. Work of Knop and Sahi [26] on Jack polynomials helped to validate this approach, and their independently derived recurrences [25], [39] ultimately inspired the combinatorial formula for nonsymmetric Macdonald polynomials of Haglund, Haiman, and Loehr [18].
Generalizing Haglund's elegant combinatorial formula for [16], [17], Haglund, Haiman and Loehr [18] gave a combinatorial formula for as where the sum is over certain positive integer fillings T of the diagram of the composition a and coinv and maj are nonnegative integer statistics. In stark contrast with the symmetric case, there are no known (nor even conjectured) positivity results for the nonsymmetric Macdonald polynomials.
Demazure [12] generalized the Weyl character formula to certain submodules, eponymously named Demazure modules, which are generated by extremal weight spaces under the action of a Borel subalgebra of a Lie algebra. The resulting Demazure characters , where , for w a Weyl group element acting on a highest weight λ, arose in connection with Schubert calculus [11], and, in type A, also form a basis of the polynomial ring. Recent work of Assaf and Searles [6] indicates that the type A Demazure characters are the most natural pull backs of Schur functions to the polynomial ring. That is to say, the combinatorics of the former stabilizes to that of the latter, Therefore, in the search for polynomial analogs of Schur positivity statements for nonsymmetric Macdonald polynomials, the natural basis for comparison is the basis of Demazure characters.
Sanderson [40] first made the connection between specializations of Macdonald polynomials and Demazure characters by using the theory of nonsymmetric Macdonald polynomials in type A to construct an affine Demazure module with graded character , parallel to the construction of Garsia and Procesi [14] for Hall-Littlewood symmetric functions . Ion [20] generalized this result to nonsymmetric Macdonald polynomials in general type using the method of intertwiners in double affine Hecke algebras to realize as a single affine Demazure character. Inspired by this, Lenart, Naito, Sagaki, Schilling and Shimozono [30] constructed a connected Kirillov–Reshetikhin crystal to give a combinatorial proof of the coincidence with affine Demazure characters using similar methods.
Recently, Assaf [1] proved the specialization is a nonnegative, graded sum of finite Demazure characters. The proof utilizes the machinery of weak dual equivalence [8]. Hence, the resulting formula is difficult to work with and, in practice, requires computing the full fundamental slide polynomial [5] expansion of . In order to have a better understanding of this nonnegativity and to have a deeper connection with the underlying representation theory of Demazure modules, we use crystal theory to give a new proof of this graded nonnegativity for finite Demazure characters from which we extract an explicit formula for the Demazure expansion. The immediate benefit of our new approach is two-fold. On the one hand, our method yields a formula which is very easily computable. On the other, weak dual equivalence exists only for the general linear group whereas the crystal theory used in our new approach extends to all types, thus our results give hope that these new methods might be a key to a result in general type.
Kashiwara [22] introduced the notion of crystal bases in his study of the representation theory of quantized universal enveloping algebras for complex, semi-simple Lie algebra at . The theory of canonical bases, developed earlier by Lusztig [32], studies the same problem from a more geometric viewpoint, though many of the main ideas from [32] carry over to [22]. A crystal base is a basis of a representation for on which the Chevalley generators have a relatively simple action. Combinatorially, a crystal is a directed, colored graph with vertex set given by the crystal base and directed edges given by deformations of the Chevalley generators. By constructing a crystal for a set of combinatorial objects, we create a combinatorial skeleton of the corresponding modules whose character is the generating function of those objects. In particular, the generating function is Schur positive. Moreover, crystal theory provides unique highest weight elements, from which tractable formulas can be derived. Stembridge [44] gave a local characterization of simply-laced crystals that allows one to prove that a given construction is indeed a crystal by analyzing local properties of the raising and lowering operators which give rise to the edges of the graph.
Demazure crystals, whose structure was conjectured by Littlemann [31] and proved by Kashiwara [23], are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. Unlike full crystals, Demazure crystals are not uniquely characterized by their highest weight elements. Further complicating matters, in the Demazure case the crystals are truncated so Stembridge's methods are not immediately applicable.
In this paper, we remedy this impediment and develop a new local characterization of Demazure crystals. These tools allow us to overcome the apparent limitations of Stembridge's axioms and readily surpass the difficulties mentioned above. In particular, in §4 we consider different families of subsets of crystals with certain nice properties. This leads us to Definition 4.15, where we present six local axioms for a subset of a normal crystal to be considered a Demazure subset. Our first principal result, stated precisely in Theorem 4.16, Theorem 4.18, is the following: Theorem Every Demazure crystal is a Demazure subset of a normal crystal, and every Demazure subset of a normal crystal is a Demazure crystal.
Furthermore, since the characters for Demazure crystals depend on the highest weight and an element of the Weyl group, the existence of an explicit Demazure crystal does not immediately yield a formula for the character. Instead, the Demazure character is determined by a specific lowest weight, but since lowest weights are not unique, this requires inspecting the entire crystal to determine the global lowest weight, which we term the Demazure lowest weight. To overcome this obstacle, we present an algorithm in Definition 4.20 that deterministically computes the Demazure lowest weight beginning with the unique highest weight. That is, from Theorem 4.22, we obtain a formula: Theorem If D is a Demazure crystal, then its character is where the latter sum is over highest weight elements, is the result of applying Definition 4.20 to b, and denotes the Demazure character.
Our motivation for deriving the results in §4 provides our immediate application, which is to construct Demazure crystals whose characters are the nonsymmetric Macdonald polynomials specialized at . This we do in §5 Definition 5.3, in which we define explicit raising and lowering operators on semistandard key tabloids, the combinatorial objects for which the specialized nonsymmetric Macdonald polynomials are the generating functions. We use Kohnert's paradigm for Demazure characters to define an explicit map that embeds our structure into the normal crystals on semistandard Young tableaux, giving Theorem 5.33: Theorem The map from semistandard key tabloids to semistandard Young tableaux is a weight-preserving injective map that intertwines the crystal operators. In particular, the image of is a subset of a normal crystal. Theorem The graph on semistandard key tabloids defined by the raising operators is a Demazure subcrystal of a normal crystal. Therefore, writing , we have where denotes the set of semistandard key tabloids of shape b, and maj is the Haglund–Haiman–Loehr statistic. In particular, nonsymmetric Macdonald polynomials specialized at are a nonnegative q-graded sum of Demazure characters.
Our results give an explicit formula for this expansion, however, in the symmetric case we can say more. The Hall–Littlewood symmetric functions may be regarded as the specialization of Macdonald symmetric functions. They are long known to be Schur positive and their Schur coefficients, the Kostka–Foulkes polynomials , have rich interpretations in geometry and representation theory. Lascoux and Schützenberger [28] recursively defined a statistic called charge on these objects that precisely gives . Using our formula for nonsymmetric Macdonald polynomials, we arrive at a new expression for using the much simpler maj statistic. In Theorem 6.5, we prove the following: Theorem The Kostka–Foulkes polynomials are given by where denotes the conjugate of λ.
Section snippets
Macdonald polynomials
Symmetric functions arise in many areas of mathematics, appearing as characters of polynomial representations of the general linear group, Frobenius characters of representations of the symmetric groups, and as natural representatives of Schubert classes for Grassmannians. In these contexts, the Schur functions and their generalizations play the pivotal role of irreducible objects, and the problem of determining the coefficients of a given symmetric function in the Schur basis combinatorializes
Crystals for the general linear group
Kashiwara's theory of crystal bases [22] provides a powerful tool for studying representations as well as for categorifying Schur positive symmetric functions by providing the combinatorial skeleton of a representation whose character is the given symmetric function.
In §3.1, we recall the basic definitions for abstract and normal crystals in the case when is the general linear group . In §3.2, we consider the action of the Borel subalgebra consisting of upper-triangular matrices on
Characterizations of Demazure crystals
One can prove that a given colored, directed graph with weighted vertices is the crystal of a -representation by finding a weight-preserving bijection with semistandard Young tableaux that intertwines the crystal operators. To circumvent this difficulty of finding an explicit bijection, Stembridge [44] gave a local characterization of normal crystals for simply-laced types that allows one to determine directly if a given colored, directed graph is the crystal for some representation.
Demazure crystal on key tabloids
We now apply the tools and techniques of crystal theory to the specialized nonsymmetric Macdonald polynomials. In particular, we will define crystal operators on semistandard key tabloids that generate a Demazure crystal, thereby giving a new combinatorial proof of Theorem 2.6 along with a tractable formula for the coefficients that arise in the Demazure expansion of a specialized nonsymmetric Macdonald polynomial.
In §5.1, we define explicit raising and lowering operators on semistandard key
Combinatorial formulas
Sanderson [40] first made the connection between specializations of Macdonald polynomials and Demazure characters by using the theory of nonsymmetric Macdonald polynomials in type A to construct an affine Demazure module with graded character , parallel to the construction of Garsia and Procesi [14] for Hall-Littlewood symmetric functions . Ion [20] generalized this result to nonsymmetric Macdonald polynomials in general type using the method of intertwiners in double affine
Acknowledgements
The authors are grateful to the referees for careful reading and invaluable feedback that greatly improved the exposition. Any remaining errors are due solely to the authors.
References (44)
An inversion statistic for reduced words
Adv. in Appl. Math.
(2019)- et al.
Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
Adv. in Math.
(2017) - et al.
Kohnert tableaux and a lifting of quasi-Schur functions
J. Combin. Theory Ser. A
(2018) - et al.
On certain graded -modules and the q-Kostka polynomials
Adv. Math.
(1992) - et al.
Crystal graphs for representations of the q-analogue of classical Lie algebras
J. Algebra
(1994) Crystal graphs and Young tableaux
J. Algebra
(1995)Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials
Trans. Amer. Math. Soc.
(2018)- et al.
Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials
- et al.
A Demazure crystal construction for Schubert polynomials
Algebraic Combinatorics
(2018) Demazure crystals for Kohnert polynomials
Weak dual equivalence for polynomials
Forum Math. Sigma
Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups
Nonsymmetric Macdonald polynomials
Internat. Math. Res. Notices
Désingularisation des variétés de Schubert généralisées
Ann. Sci. École Norm. Sup. (4)
Une nouvelle formule des caractères
Bull. Sci. Math. (2)
Some natural bigraded -modules and -Kostka coefficients
Electron. J. Combin.
A graded representation model for Macdonald's polynomials
Proc. Nat. Acad. Sci. U.S.A.
A combinatorial model for the Macdonald polynomials
Proc. Natl. Acad. Sci. USA
A combinatorial formula for Macdonald polynomials
J. Amer. Math. Soc.
A combinatorial formula for nonsymmetric Macdonald polynomials
Amer. J. Math.
Hilbert schemes, polygraphs and the Macdonald positivity conjecture
J. Amer. Math. Soc.
Nonsymmetric Macdonald polynomials and Demazure characters
Duke Math. J.
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S.A. supported in part by NSF DMS-1763336. N.G. was supported in part by NSF grants DMS-1255334 and DMS-1664240.