Demazure crystals for specialized nonsymmetric Macdonald polynomials

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Abstract

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.

Introduction

Macdonald [33] defined symmetric functions Pμ(X;q,t) with two parameters q,t indexed by partitions as the unique symmetric function basis satisfying certain triangularity (with respect to monomials in infinitely many variables X=x1,x2,) and orthogonality (with respect to a generalized Hall inner product) conditions. The Macdonald symmetric functions give a simultaneous generalization of Hall–Littlewood symmetric functions Pλ(X;0,t) and Jack symmetric functions limt1Pλ(X;tα,t).

The coefficients of Pμ(X;q,t) when written as a sum of monomials are rational functions in the parameters q and t. Macdonald conjectured that the Kostka–Macdonald coefficients Kλ,μ(q,t) defined by expanding the integral form Jμ(X;q,t), a scalar multiple of the original Pμ(X;q,t), into the plethystic Schur basis,Jμ(X;q,t)=λKλ,μ(q,t)sλ[X(1t)], are polynomials in q and t with nonnegative integer coefficients. Here the square brackets denote plethystic substitution. In short, sλ[A] means sλ 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 isHμ(X;q,t)=Jμ[X/(1t);q,t]. Thus, the Kostka–Macdonald coefficients give the Schur function expansion of Hμ(X;q,t). 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 Ea(X;q,t) are indexed by weak compositions and form a basis for the full polynomial ring. They generalize Macdonald polynomials in the sense thatE0m×a(x1,,xm,0,,0;q,t)=Psort(a)(x1,,xm;q,t), where 0m×a 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 Hμ(X;q,t) [16], [17], Haglund, Haiman and Loehr [18] gave a combinatorial formula for Ea(X;q,t) asEa(X;q,t)=T:a[n]nonattackingqmaj(T)tcoinv(T)Xwt(T)cleft(c)1t1qleg(c)+1tarm(c)+1, 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 κa, where a=wλ, 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,κ0m×a(x1,,xm,0,,0)=ssort(a)(x1,,xm). 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 Pμ(X;q,0), parallel to the construction of Garsia and Procesi [14] for Hall-Littlewood symmetric functions Hμ(X;0,t). Ion [20] generalized this result to nonsymmetric Macdonald polynomials in general type using the method of intertwiners in double affine Hecke algebras to realize Ea(X;q,0) 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 Ea(X;q,0) 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 Ea(X;q,0). 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 Uq(g) for complex, semi-simple Lie algebra g at q=0. 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 Uq(g) 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 gln crystal for a set of combinatorial objects, we create a combinatorial skeleton of the corresponding gln 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 gln 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 gln crystal is a Demazure subset of a normal gln crystal, and every Demazure subset of a normal gln crystal is a Demazure gln crystal.

This provides a universal method for proving that a given subset of a 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 gln crystal, then its character isch(D)=bDx1wt(b)1x2wt(b)2xnwt(b)n=bDbhighest weightκwt(Z(b)), where the latter sum is over highest weight elements, Z(b) is the result of applying Definition 4.20 to b, and κa denotes the Demazure character.

Thus, we have an efficient formula for characters of Demazure crystals.

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 t=0. 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 gln 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 gln crystal.

Hence, we are now in the situation to apply our characterization of Demazure crystals, which culminates in Theorem 5.42, and states:

Theorem

The graph on semistandard key tabloids defined by the raising operators is a Demazure subcrystal of a normal crystal. Therefore, writing Eb(Xn;q,0)=aKa,b(q)κa(Xn), we haveKa,b(q)=TSSKD(b)Thighest weightwt(Z(T))=aqmaj(T), where SSKD(b) 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 t=0 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 q=0 specialization of Macdonald symmetric functions. They are long known to be Schur positive and their Schur coefficients, the Kostka–Foulkes polynomials Kλ,μ(t), have rich interpretations in geometry and representation theory. Lascoux and Schützenberger [28] recursively defined a statistic called charge on these objects that precisely gives Kλ,μ(t). Using our formula for nonsymmetric Macdonald polynomials, we arrive at a new expression for Kλ,μ(t) using the much simpler maj statistic. In Theorem 6.5, we prove the following:

Theorem

The Kostka–Foulkes polynomials Kλ,μ(t) are given byKλ,μ(t)=TSSKD(0|μ|μ1×rev(μ))Thighest weightwt(T)=λtmaj(T), where λ denotes the conjugate of λ.

We conclude by noting that the Demazure coefficients of specialized nonsymmetric Macdonald polynomials give a refinement of the Kostka–Foulkes polynomials that removes certain multiplicities. Moreover, as nonnegative expansions into Demazure characters are becoming more ubiquitous among geometrically significant bases for the polynomial ring, we expect our methods to have wider applications to come.

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 g is the general linear group gln. In §3.2, we consider the action of the Borel subalgebra b 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 gln-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 Pμ(X;q,0), parallel to the construction of Garsia and Procesi [14] for Hall-Littlewood symmetric functions Hμ(X;0,t). 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.

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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.

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