David B Rush

Cyclic Sieving and Plethysm Coefficients

Keywords: plethysms, Schützenberger involution, jeu-de-taquin promotion, canonical bases, Kashiwara crystals, cyclic sieving phenomenon

A combinatorial expression for the coefficient of the Schur function s λ in the expansion of the plethysm p d n/d • sµ is given for all d dividing n for the cases in which n = 2 or λ is rectangular. In these cases, the coefficient p d n/d •sµ, s λ is shown to count, up to sign, the number of fixed points of an s n µ , s λ -element set under the d th power of an order n cyclic action. If n = 2, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion.

This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case n = 2 is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Résumé. Une expression combinatoire pour le coefficient de la fonction de Schur s λ dans l'expansion du pléthysme p d n/d •sµ est donné pour tous d que disent n, dans les cas où n = 2, ou λ est rectangulaire. Dans ces cas, le coefficient p d n/d • sµ, s λ se montre à compter, où l'on ignore le signe, le nombre des point fixés d'un ensemble de s n µ , s λ éléments sous la puissance d e d'une action cyclique de l'ordre n. Si n = 2, l'action est l'involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si λ est rectangulaire, l'action est une certaine puissance de l'avancement jeu-de-taquin de Schützenberger et Shimozono. Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas n = 2, la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d'une fonction de Schur.

Introduction

A principal concern of algebraic combinatorics is the identification of collections of combinatorial objects that occur in algebraically significant multiplicities. Perhaps the most celebrated success in this endeavor is the Littlewood-Richardson rule, which gives a combinatorial description for the coefficient of each Schur function arising in the expansion of a product of Schur functions on the Schur basis. For the case of a Schur function s µ raised to the n th power, there is a natural order n cyclic action on the objects specified by the Littlewood-Richardson rule for the coefficient of s λ , provided that n = 2 or λ is rectangular. In 2, . . . , (k + 1)m} for all 0 ≤ k ≤ n -1. Let j act on PYTab(λ, µ n ) by m iterations of jeu-de-taquin promotion. Then, for all positive integers d dividing n, |{T ∈ PYTab(λ, µ n ) : j d (T ) = T }| = ± p d n/d • s µ , s λ .

From Theorems 3.1 and 3.2 in [START_REF] Lascoux | Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties[END_REF], we see that the Hall-Littlewood symmetric function Q 1 n (q) specializes (up to sign) at q = e 2πi n to p gcd(n, )

n/gcd(n, ) . Therefore, we may interpret Theorem 1.2 as analogous to exhibiting an instance of the cyclic sieving phenomenon.

Corollary 1.3. Let λ be a rectangular partition. Let j act on PYTab(λ, µ n ) by m iterations of jeu-detaquin promotion. Then, for all integers ,

|{T ∈ PYTab(λ, µ n ) : j (T ) = T }| = ± Q 1 n (e 2πi n ) • s µ , s λ .
Remark 1.4. The signs appearing in Theorems 1.1 and 1.2 are predictable, and depend upon λ, d, and n only. Consult section 4 (or [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF] itself), for more details. [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF] is by no means the first attempt at computing the coefficients in the expansions of power-sum plethysms. In [START_REF] Carré | Splitting the square of a Schur function into its symmetric and antisymmetric parts[END_REF], a rule for splitting the square of a Schur function into its symmetric and antisymmetric parts was devised, the crux of which was a demonstration that the coefficient p 2 • s µ , s λ counted, up to sign, the number of domino tableaux of shape λ and content µ with Yamanouchi reading words. Two years later, [START_REF] Lascoux | Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties[END_REF] introduced a new family of symmetric functions, today referred to as LLT functions, and proposed that the plethysm p d n/d • s µ could be expressed as the specialization of an LLT function at an appropriate root of unity (as indeed p d n/d is the specialization of a Hall-Littlewood function). However, the Lascoux-Leclerc-Thibon conjecture remains unproven, and the Carré-Leclerc rule has not been generalized to cases beyond n = 2, for the concept of Yamanouchi reading words has not been extended to n-ribbon tableaux for n ≥ 3.

Thus, Theorem 1.1 does not give the first combinatorial expression for the coefficient p 2 • s µ , s λ , but it distinguishes itself from the existing Carré-Leclerc formula by its natural compatibility with the Littlewood-Richardson rule. It is sufficiently robust that the techniques involved in its derivation are applicable to a whole class of plethysm coefficients with n > 2, addressed in Theorem 1.2, which is new in content and in form. Furthermore, the author has shown in unpublished work that a bijection of [START_REF] Berenstein | Domino tableaux, Schützenberger involution, and the symmetric group action[END_REF] between domino tableaux and tableaux stable under evacuation restricts to a bijection between those tableaux specified in the Carré-Leclerc rule and in Theorem 1.1, respectively. It follows that Theorem 1.1 actually recovers the Carré-Leclerc result.

To prove Theorems 1.1 and 1.2, we turn to the theory of Lusztig canonical bases, which provides an algebraic setting for the Schützenberger actions evacuation and promotion. In particular, we consider an irreducible representation of GL mn (C) for which there exists a basis indexed by the semistandard tableaux of shape λ with entries in {1, 2, . . . , mn} such that, if n = 2, the long element w 0 ∈ S mn → GL mn permutes the basis elements (up to sign) by evacuation, and, if λ is rectangular, the long cycle c mn ∈ S mn → GL mn permutes the basis elements (up to sign) by promotion.

With a suitable basis in hand, we proceed to compute the character of the representation at a particular element of GL mn . If n = 2, we compute where the block diag(y 1 , y 2 , . . . , y d ) occurs n/d times along the main diagonal, and y i in turn represents the block diag(y i,1 , y i,2 , . . . , y i,m ) for all 1 ≤ i ≤ d. These character evaluations pick out the fixed points of the relevant order n cyclic actions. Furthermore, they may be calculated by diagonalization of the indicated elements, for characters are class functions, and the values of the irreducible characters of GL mn at diagonal matrices are well known. A careful inspection of the resulting formulae yields the desired identities.

χ(w 0 • diag(x 1 , x 2 , . . . , x m , x m , . . . , x 2 , x 1 )),
The relationship between w 0 and evacuation was first discovered in [START_REF] Berenstein | Canonical bases for the quantum group of type A r and piecewise-linear combinatorics[END_REF], in the context of a basis dual to Lusztig's canonical basis. Herein we opt for an essentially equivalent basis constructed in [START_REF] Skandera | On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations[END_REF], which was used by Rhoades to detect the analogous relationship between c mn and promotion. From the observations that w 0 and c mn lift the actions of evacuation and promotion, respectively, with respect to the dual canonical basis (or something like it), [START_REF] Stembridge | Canonical bases and self-evacuating tableaux[END_REF] and [START_REF] Rhoades | Cyclic sieving, promotion, and representation theory[END_REF] deduced correspondences between fixed points of Schützenberger actions and ribbon tableaux. These theorems inspired our results.

Theorem 1.5 ([18], Corollary 4.2). Let Tab(λ, µµ) be the set of all semistandard tableaux of shape λ and content µµ, and let ξ act on Tab(λ, µµ) by the Schützenberger involution. Then

|{T ∈ Tab(λ, µµ) : ξ(T ) = T }|
is the number of domino tableaux of shape λ and content µ.

Theorem 1.6 ( [START_REF] Rhoades | Cyclic sieving, promotion, and representation theory[END_REF], proof of Theorem 1.5). Let λ be a rectangular partition, and let Tab(λ, µ n ) be the set of all semistandard tableaux of shape λ and content µ n . Let j act on Tab(λ, µ n ) by m iterations of jeu-de-taquin promotion. Then, for all positive integers d dividing n,

|{T ∈ Tab(λ, µ n ) : j d (T ) = T }|
is the number of (n/d)-ribbon tableaux of shape λ and content µ d .

Unfortunately, the proofs of Theorems 1.5 and 1.6 cannot be directly adapted to obtain Theorems 1.1 and 1.2. In order for the Yamanouchi restrictions on our tableaux sets to be made to appear in our character evaluations, an additional point of subtlety is needed. We find relief in the insights offered us by the theory of Kashiwara crystals, which provides a framework not only for the study of the Schützenberger actions, but also for the reformulation of the Yamanouchi restrictions in terms of natural operators on semistandard tableaux.

Let g be a complex semisimple Lie algebra with weight lattice W , and choose a set of simple roots ∆ = {α 1 , α 2 , . . . , α t }. A g-crystal is a finite set B equipped with a weight map wt : B → W and a pair of raising and lowering operators e i , f i :

B → B {0} for each i that obey certain conditions. Most notably, for all b ∈ B, if e i • b is nonzero, then wt(e i • b) = wt(b) + α i , and if f i • b is nonzero, then wt(f i • b) = wt(b) -α i .
If g = sl mn , then W is a quotient of Z mn , and we may choose for our simple roots the images of the vectors i -i+1 for 1 ≤ i ≤ mn-1, where i denotes the i th standard basis vector for all 1 ≤ i ≤ mn. In this case, we may take B to be the set of semistandard tableaux of shape λ with entries in {1, 2, . . . , mn}, with the weight of each tableau encoded in its content. As we see in section 3, there exists a suitable choice of operators e i and f i so that B assumes the structure of a g-crystal, and that the word of a tableau b ∈ B is Yamanouchi with respect to the letters i and i + 1 if and only if e i vanishes at b, and anti-Yamanouchi with respect to i and i + 1 if and only if f i vanishes at b. Furthermore, evacuation and promotion act on the set of crystal operators by conjugation (essentially), which explains why they act on the tableaux sets indicated in our main theorems.

We close the introduction with an outline of the rest of the article. In section 2, we introduce plethysms, and we recall the observation of [START_REF] Lascoux | Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties[END_REF] that the classical relationship between tableaux and Schur functions evinces a more general relationship between ribbon tableaux and power-sum plethysms of Schur functions. In section 3, we define Kashiwara crystals for a complex semisimple Lie algebra, before specializing to the sl mn setting, where we show how to assign a crystal structure to the pertinent tableaux sets. We also examine how the Schützenberger actions interact with the raising and lowering crystal operators. Finally, in section 4, we sketch proofs of Theorems 1.1 and 1.2.

Background on Plethysms

In this extended abstract, we assume familiarity with the basic facts about Young tableaux and symmetric functions. (Accounts of the fundamentals can be found in [START_REF] Fulton | Young tableaux[END_REF], Chapters 1-6; for a treatment specific to our needs, see [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF].) We start, then, with the rudiments of plethysms, following [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF].

Definition 2.1. Let f, g ∈ Λ, and let g be written as a sum of monomials, so that g = η u η x η , where η ranges over an infinite set of compositions. Let {y i } ∞ i=1 be a collection of proxy variables defined by

∞ i=1 (1 + y i t) = η (1 + x η t) uη .
The plethysm of f and g, which we denote by f • g, is the symmetric function f (y 1 , y 2 , . . .).

Remark 2.2. Although the relation

∞ i=1 (1 + y i t) = η (1 + x η t) uη
only determines the elementary symmetric functions in the variables y 1 , y 2 , . . . (viz., e 1 (y 1 , y 2 , . . .) = y 1 + y 2 + • • • , e 2 (y 1 , y 2 , . . .) = y 1 y 2 + y 1 y 3 + y 2 y 3 + • • • , etc.), it is well known that the ring of symmetric functions is generated as a Z-algebra by the elementary symmetric functions, so the plethysm f • g = f (y 1 , y 2 , . . .) is indeed well-defined.

The following observation follows immediately from Definition 2.1.

Proposition 2.3. For all f ∈ Λ, the map Λ → Λ given by g → g • f is a ring homomorphism.

There exists a family of symmetric functions for which the other choice of map given by plethysm, i.e. g → f • g, is also a ring homomorphism, for all f belonging to this family. Definition 2.4. For all positive integers k, the k th power-sum symmetric function in the variables x 1 , x 2 , . . .

is p k := x k 1 + x k 2 + • • • . Proposition 2.5. Let g ∈ Λ, and let k be a positive integer. Then p k • g = g • p k = g(x k
1 , x k 2 , . . .). We may conclude that the map given by g → p k • g is a ring homomorphism for all positive integers k. We are therefore permitted to introduce an adjoint operator, which we denote by ϕ k , given by f → κ f, p k •s κ s κ , where the sum ranges over all partitions κ. Clearly, the equality ϕ k (f ), g = f, p k •g holds for all f, g ∈ Λ, which explains the nomenclature.

Let κ be a partition. Just as the ordinary tableaux of shape κ index the monomials of the Schur function s κ , the k-ribbon tableaux of shape κ index the monomials of the symmetric function ϕ k (s κ ).

Theorem 2.6. Let κ be a partition, and suppose that the k-core of κ is empty. For all compositions η of |κ| k , we denote the monomial x η1 1 x η2 2 • • • by x η , and, for all k-ribbon tableaux T of shape κ and content η, we write x T for x η . Then ϕ k (s κ ) = k (κ) T x T , where the sum ranges over all k-ribbon tableaux of shape κ, and k (κ) denotes the k-sign of κ.

Proof. Let κ (1) , κ (2) , . . . , κ (k) be the k-quotient of κ. Since the k-core of κ is empty, it follows from a result of [START_REF] Littlewood | Modular representations of symmetric groups[END_REF]

that ϕ k (s κ ) = k (κ)s κ (1) s κ (2) • • • s κ (k)
. However, from Equation 24 in [START_REF] Lascoux | Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties[END_REF], we see that

s κ (1) s κ (2) • • • s κ (k) = T x T ,
where the sum ranges over all k-ribbon tableaux of shape κ, as desired.

In view of Theorem 2.6, it is natural to ask if there is an analogue of the Littlewood-Richardson rule that describes the expansion coefficients of the power-sum plethysms p n •s µ , or, more generally, p d n/d •s µ , for d dividing n. In the following sections, we outline how [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF] provides a partial affirmative answer.

Crystal Structure on Tableaux

For a complex semisimple Lie algebra g, Kashiwara's g-crystals constitute a class of combinatorial models patterned on representations of g. If g is simply laced, there exists a set of axioms, enumerated in [START_REF] Stembridge | A local characterization of simply-laced crystals[END_REF], that characterize the crystals arising directly from g-representations, which he calls regular. Given a partition κ with at most s positive parts, we may consider κ as a partition with s parts. The combinatorics of the weight space decomposition of the irreducible sl s -representation with highest weight encoded in κ is captured in the regular sl s -crystal structure assigned to the semistandard tableaux of shape κ with entries in {1, 2, . . . , s}.

In this section, we review the crystal structure on tableaux, and we observe that it offers a natural setting for the consideration of evacuation and promotion, due to the relationship between these actions and the raising and lowering crystal operators. We also see that the crystal perspective facilitates a recasting of the Yamanouchi conditions on tableaux reading words in terms of the vanishing or nonvanishing of the raising and lowering operators at the corresponding tableaux, viewed as crystal elements. We begin with the definition of a crystal, following [START_REF] Joseph | Quantum groups and their primitive ideals[END_REF]. Definition 3.1. Let g be a complex semisimple Lie algebra with weight lattice W . Let {α 1 , α 2 , . . . , α t } be a choice of simple roots, and let {α ∨ 1 , α ∨ 2 , . . . , α ∨ t } be the corresponding simple coroots. A g-crystal is a finite set B equipped with a map wt : B → W and a pair of operators e i , f i : B → B {0} for each 1 ≤ i ≤ t that satisfy the following conditions:

(i) max{ : f i • b = 0} -max{ : e i • b = 0} = wt(b), α ∨ i for all b ∈ B; (ii) e i • b = 0 implies wt(e i • b) = wt(b) + α i and f i • b = 0 implies wt(f i • b) = wt(b) -α i for all b ∈ B; (iii) b = e i • b if and only if b = f i • b for all b, b ∈ B.
We refer to e i as the raising operator associated to α i , and we refer to f i as the lowering operator associated to α i . We write i (b) := max{ : e i • b = 0} for the maximum number of times the raising operator e i may be applied to b without vanishing, and we write φ i (b) := max{ : f i • b = 0} for the maximum number of times the lowering operator f i may be applied to b without vanishing. If a g-crystal B satisfies the additional conditions (P4), (P5), (P6), (P5'), and (P6') of [START_REF] Stembridge | A local characterization of simply-laced crystals[END_REF], we say that B is regular. Definition 3.2. Let B and B be g-crystals. A map of sets π : B → B is a morphism of crystals if πe i = e i π and πf i = f i π for all 1 ≤ i ≤ t. (Here we tacitly stipulate π(0) := 0.) If π is bijective, we say π is an isomorphism. Specializing to the case g = sl s , we take as our Cartan subalgebra h the subspace of traceless diagonal matrices, and we identify h * with the quotient space C s /( 1 + 2 + • • • + s )C, where i denotes the i th standard basis vector for all 1 ≤ i ≤ s. Writing E i for the image of i in h * for all i, we note that the weight lattice W is generated over Z by {E 1 , E 2 , . . . , E s }, and we choose the set of simple roots {α 1 , α 2 , . . . , α s-1 } in accordance with the rule α i := E i -E i+1 for all 1 ≤ i ≤ s -1.

Proposition 3.8 ([6]

). Let κ and ι be partitions, each with s parts, such that ι i ≤ κ i for all positive parts ι i of ι. Let B κ/ι be the set of semistandard skew tableaux of shape κ/ι with entries in {1, 2, . . . , s}.

Let the maps

wt : B κ/ι → Z s /( 1 + 2 + • • • + s )Z h i,j , k i,j : B κ/ι → Z e i , f i : B κ/ι → B κ/ι {0}
be given for all 1 ≤ i ≤ s -1 and j ∈ N by stipulating, for all T ∈ B κ/ι :

• wt(T ) to be the image in

Z s /( 1 + 2 + • • • + n )Z of the content of T ;
• h i,j (T ) to be the number of occurrences of i + 1 in the j th column of T or to the right minus the number of occurrences of i in the j th column of T or to the right;

• k i,j (T ) to be the number of occurrences of i in the j th column of T or to the left minus the number of occurrences of i + 1 in the j th column or to the left;

• e i (T ) to be the skew tableau with an i in place of an i + 1 in the rightmost column for which h i,j (T ) is maximal and positive if such a column exists, and 0 otherwise;

• f i (T ) to be the skew tableau with an i + 1 in place of an i in the leftmost column for which k i,j (T ) is maximal and positive if such a column exists, and 0 otherwise.

Then the set B κ/ι equipped with the map wt and the operators e i , f i for all 1 ≤ i ≤ s-1 is an sl s -crystal.

Proposition 3.9 ( [START_REF] Stembridge | A local characterization of simply-laced crystals[END_REF]). Let κ be a partition with s parts. The sl s -crystal B κ := B κ/∅ defined in Proposition 3.8 is a regular connected crystal of highest weight κ

+ ( 1 + 2 + • • • + s )Z.
The highest weight element is the unique tableau of shape κ and content κ.

We now observe that Schützenberger's jeu de taquin respects the crystal structure on tableaux in the sense that jeu-de-taquin slides commute with the raising and lowering operators. Since evacuation and promotion may be defined via jeu de taquin (cf. [START_REF] Schützenberger | Promotion des morphismes d'ensembles ordonnés[END_REF] and [START_REF] Shimozono | Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties[END_REF]), it should be no surprise that they inherit this compatibility. Proposition 3.10 ( [START_REF] Lascoux | Keys & standard bases[END_REF]). Let κ and ι be nonempty partitions such that ι i ≤ κ i for all positive parts ι i of ι, and let C be an inside corner of κ/ι. For all semistandard skew tableaux T of shape κ/ι, let jdt(T ) be the result of a jeu-de-taquin slide on T starting from C, and set jdt(0) := 0. Then e i • jdt(T ) = jdt(e i • T ) and

f i • jdt(T ) = jdt(f i • T ) for all T ∈ B κ/ι and 1 ≤ i ≤ s -1.
Proposition 3.11 ([7]). Let κ be a partition with s parts, and let ξ : B κ → B κ be the Schützenberger involution. Set ξ(0) := 0. Then, for all T ∈ B κ :

(i) wt(ξ(T )) = w 0 • wt(T ); (ii) ξ(e i • T ) = f s-i • ξ(T ) and ξ(f i • T ) = e s-i • ξ(T ) for all 1 ≤ i ≤ s -1.
Proposition 3.12 ( [START_REF] Shimozono | Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties[END_REF]). Let κ be a partition with s parts, and let pr : B κ → B κ be jeu-de-taquin promotion. Set pr(0) := 0. Then, for all T ∈ B κ :

(i) wt(pr(T )) = c s • wt(T ); (ii) pr(e i • T ) = e i+1 • pr(T ) and pr(f i • T ) = f i+1 • pr(T ) for all 1 ≤ i ≤ s -1.
The following theorem reveals the reason why we restrict our attention to rectangular partitions in the statement of Theorem 1.2. Theorem 3.13 ([16]). Let κ be a partition with s parts, and let pr : B κ → B κ be jeu-de-taquin promotion. Then pr s acts as the identity if and only if κ is rectangular.

To close the section, as promised, we reinterpret the Yamanouchi conditions on reading words as vanishing conditions on crystal operators. Proposition 3.14. Let κ be a partition with s parts, and let T be a tableau of shape κ. For all 1 ≤ i < i ≤ s -1, the word of T is Yamanouchi (anti-Yamanouchi) in the subset {i, i + 1, . . . , i } if and only if the raising operators e i , e i+1 , . . . , e i -1 (lowering operators f i , f i+1 , . . . , f i -1 ) all vanish at T .

Proofs of Theorems 1.1 and 1.2

In this section, we sketch proofs of our main theorems. We start by delineating the properties of the basis of Kazhdan-Lusztig immanants constructed in [START_REF] Skandera | On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations[END_REF]. For κ a partition with at most s positive parts, we consider κ as a partition with s parts. We note that the action of the long element w 0 ∈ S s ⊂ GL s (C) on the immanants generating a GL s (C)-representation associated to κ lifts (up to sign) the Schützenberger involution on the tableaux in the sl s -crystal B κ , and, analogously, that the action of the long cycle c s ∈ S s ⊂ GL s (C) on immanants lifts (up to sign) jeu-de-taquin promotion if κ is rectangular. Theorem 4.1 ([13], [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF]). Let κ be a partition of t with s parts, and let V κ,s be the dual of the irreducible polynomial GL s (C)-representation with highest weight κ. For all compositions η of t with s parts and semistandard tableaux U of shape κ and content η, let I η (U ) ∈ V s be the Kazhdan-Lusztig immanant associated to η and U . Set I η := {I η (U ) : U is a semistandard tableau of shape κ and content η}.

Then the following claims hold.

(i) The set η I η , where η ranges over all compositions of t with s parts, constitutes a basis for V κ,s .

(ii) For all compositions η of t with s parts, the set I η constitutes a basis for the weight space of V T,s corresponding to the weight -η, which we denote by V κ,s,η .

(iii) Let w 0 be the long element in S s , and let ξ be the Schützenberger involution. Then

w 0 • I η (U ) = (-1) v(κ) • I w0•η (ξ(U )).
(iv) Let c s be the long cycle in S s , and let pr be jeu-de-taquin promotion. If κ is rectangular with exactly a positive parts, then

c s • I η (U ) = (-1) ηs(a-1) I cs•η (pr(U )).

Proof of Theorem 1.1

Let λ be a partition with 2m parts. Suppose that 2 divides |λ|, and let µ = (µ 1 , µ 2 , . . . , µ m ) be a partition of |λ|/2. Let B λ be the set of semistandard tableaux of shape λ, endowed with an sl 2m -crystal structure by the sets of raising and lowering operators {e i } 2m-1 i=1 and {f i } 2m-1 i=1 , respectively. The key to our proof is the assignment of an (sl m ⊕ sl m )-crystal structure to B λ that allows us to inspect the action of ξ on its connected components. This process provides a combinatorial model for the decomposition into irreducible components of the restriction to GL m (C) × GL m (C) of the irreducible GL 2m (C)-representation with highest weight λ, which underlies our character evaluation.

Recall that we chose {E 1 -E 2 , E 2 -E 3 , . . . , E 2m-1 -E 2m } as the set of simple roots for sl 2m . Here we choose {E

2 -E 1 , E 3 -E 2 , . . . , E m -E m-1 , E m+1 -E m+2 , E m+2 -E m+3 , . . . , E 2m-1 -E 2m }
as the set of simple roots for sl m ⊕ sl m . Proposition 4.2. The set B λ equipped with the map wt, the set of raising operators {f i } m-1 i=1 ∪{e i+m } m-1 i=1 , and the set of lowering operators {e i } m-1 i=1 ∪ {f i+m } m-1 i=1 is a regular (sl m ⊕ sl m )-crystal. Proposition 4.3. Let β and γ be partitions, each with m parts. Equip the set B (β,γ) := B β × B γ with the map wt × wt. For all 1 ≤ i ≤ m -1, let e i and f i act as the sl m -crystal operators e i and f i , respectively, on B β and as the identity on B γ . For all m + 1 ≤ i ≤ 2m -1, let e i and f i act as the identity on B β and as the sl m -crystal operators e i-m and f i-m , respectively, on B γ . Then B (β,γ) , together with the set of raising operators {f 1 , f 2 , . . . , f m-1 , e m+1 , e m+2 , . . . , e 2m-1 } and the set of lowering operators {e 1 , e 2 , . . . , e m-1 , f m+1 , f m+2 , . . . , f 2m-1 }, is a regular connected (sl m ⊕ sl m )-crystal. Identifying the coefficients of s µ (x -2 1 , x -2 2 , . . . , x -2 m ) in our two expressions, we may conclude that

|EYTab ξ (λ, µµ)| = 2 (λ) • s λ , p 2 • s µ .

Proof of Theorem 1.2

Let λ be a rectangular partition with mn parts. Suppose that n divides |λ|, and let µ = (µ 1 , µ 2 , . . . , µ m ) be a partition of |λ|/n. Let B λ be the set of semistandard tableaux of shape λ, endowed with a sl mn -crystal

Definition 3 . 3 .

 33 A g-crystal B is connected if the underlying graph, in which elements b and b are joined by an edge if there exists i such that e i • b = b or e i • b = b, is connected. We refer to the connected components of the underlying graph as the connected components of B. Remark 3.4. Regular connected g-crystals should be thought of in analogy with irreducible representations of g. Definition 3.5. Let B be a g-crystal. An element b ∈ B is a highest weight element if e i vanishes at b for all i. If b is the unique highest weight element of B, then B is a highest weight crystal of highest weight wt(b). Proposition 3.6 ([19]). Let g be simply laced, and let B be a regular connected g-crystal. Then B is a highest weight crystal. Proposition 3.7 ([19]). Let B and B be regular connected g-crystals with highest weight elements b and b , respectively. If φ i (b) = φ i (b ) for all 1 ≤ i ≤ t, then B and B are isomorphic.

Theorem 4 . 4 . 2 • 2 ,= 2

 44222 Let C be a connected component of the (sl m ⊕ sl m )-crystal B λ . Let b be the unique highest weight element of C. Then there exist partitions β and γ, each with m parts, such that b is of content βγ, and C is isomorphic to B (β,γ) . Corollary 4.5. Let C be a connected component of the (sl m ⊕ sl m )-crystal B λ , and let b be the unique highest weight element of C. If ξ(b) = b, then {T ∈ C : ξ(T ) = T } is empty. Otherwise, there exists a partition β = (β 1 , β 2 , . . . , β m ) such that b is of content ββ, and the isomorphism of crystals C ∼ -→ B (β,β) restricts to a bijection of sets{T ∈ C : ξ(T ) = T } ∼ -→ {(U, U ) ∈ B (β,β) : ξ(U ) = U }.We proceed to the proof of Theorem 1.1 itself. Taking characters in the GL 2m (C)-representation V λ,2m , we note thatχ(w 0 • diag(x 1 , x 2 , . . . , x m , x m , . . . , x 2 , x 1 )) = (-1) v(λ) • T ∈B λ :ξ(T )=T • • x -2Tm m = (-1) v(λ) • θ |λ|/2 |EYTab ξ (λ, θθ)| • s θ (x -2 1 , x -2 2 , . . . , x -2 m ),where the first equality follows from Theorems 4.1, and the second equality follows from Corollary 4.5. Since χ : GL 2m (C) → C is a class function, we see also thatχ(w 0 • diag(x 1 , x 2 , . . . , x m , x m , . . . , x 2 , x 1 )) = χ(diag(x 1 , x 2 , . . . , x m , -x m , . . . , -x 2 , -x 1 )) = s λ (x -1 1 , x -1 2 , . . . , x -1 m , -x -1 m , . . . , -x -1 ranges over all semistandard domino tableaux of shape λ with entries in {1, 2, . . . , m}. (Here the second equality follows from Theorem 4.1, and the third from Remark 3.2 of [18](λ) • φ 2 (s λ )(x -2 1 , x -2 2 , . . . , x -2 m ) = 2 (λ) • θ |λ|/2 s λ , p 2 • s θ s θ (x -2 1 , x -2 2 , . . . , x -2 m ).

  and, if λ is rectangular, we compute χ(c md mn • diag(y 1 , y 2 , . . . , y d , y 1 , y 2 , . . . , y d , . . . , y 1 , y 2 , . . . , y d )),

Acknowledgments

This research was undertaken at the University of Michigan, Ann Arbor, under the direction of David Speyer and with the financial support of the US National Science Foundation via grant DMS-1006294. It is the author's pleasure to extend his gratitude first and foremost to Prof. Speyer for his dedicated mentorship throughout the course of this project. He would also like to thank Michael Zieve for his leadership of the REU (Research Experiences for Undergraduates) program hosted by the University of Michigan as well as Victor Reiner, Brendon Rhoades, Richard Stanley, and John Stembridge for helpful conversations. Finally, thanks are due to Christopher Gellert for assistance in translating the abstract into the French.

The author is presently supported by the NSF Graduate Research Fellowship Program.

1365-8050

structure by the sets of raising and lowering operators {e i } mn-1 i=1 and {f i } mn-1 i=1 , respectively. The key to our proof is the assignment of an (sl m ⊕n )-crystal structure to B λ that allows us to inspect the action of j on its connected components. This process provides a combinatorial model for the decomposition into irreducible components of the restriction to GL m (C) ×n of the irreducible GL mn (C)-representation with highest weight λ, which underlies our character evaluation.

Recall that we chose {E 1 -E 2 , E 2 -E 3 , . . . , E mn-1 -E mn } as the set of simple roots for sl mn . Here we choose The remainder of the proof of Theorem 1.2 follows from a character evaluation in the GL mn (C)representation V λ,mn . In the interest of brevity, we suppress the details, but they can be found in [START_REF] David | Cyclic sieving and plethysm coefficients[END_REF].


Takeshi Ikeda

Tomoo Matsumura
email: tomoomatsumura@kaist.ac.kr

Equivariant Giambelli formula for the symplectic Grassmannians -Pfaffian Sum Formula

Keywords: symplectic Grassmannian, Schubert calculus, Giambelli formula, Pfaffian, signed permutation, k-strict partition

We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space.

Résumé. On démontre une formule close explicite, écrite comme une somme de Pfaffiens, qui décrit toute classe de Shubert équivariante pour la Grassmannienne des sous-espaces isotropes dans un espace vectoriel symplectique.

Introduction

The classical Giambelli formula expresses a general Schubert class of the Grassmannian as the determinant of a matrix whose entries are the so-called special Schubert classes. A special Schubert class is defined by the locus of subspaces having excess intersection with a fixed linear subspace. These classes also coincide with the Chern classes of the universal quotient bundle over the Grassmannian. Various extensions of the formula have been obtained. The Giambelli problem consists in finding a "closed formula" for a Schubert class in terms of those special classes, and in the torus equivariant setting it is closely related to the theory of degeneracy loci of vector bundles (cf. [START_REF] Fulton | Schubert varieties and degeneracy loci[END_REF], [Tam] and the references therein).

For the symplectic or orthogonal Grassmannians, there is a natural notion of special Schubert classes, which takes into account the isotropic conditions arising from the symplectic or orthogonal form. For the Grassmannian of maximal isotropic subspaces, the Giambelli formula, first found by Pragacz [START_REF] Pragacz | Algebro-geometric applications of Schur Sand Q-polynomials[END_REF], expresses a general Schubert class as a Pfaffian whose entries are appropriate quadratic polynomials in the special Schubert classes. Its natural equivariant version was obtained by Kazarian [Kaz00] in the context of degeneracy loci, and later it was also proved in [START_REF] Ikeda | Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian[END_REF] and [START_REF] Ikeda | Excited Young diagrams and equivariant Schubert calculus[END_REF] in terms of equivariant cohomology by using more algebraic methods. Similarly to Pragacz's formula, also Kazarian's expression is given as a single Pfaffian. On the other hand, for the non-maximal isotropic Grassmannians, an answer to the (non-equivariant) Giambelli problem was given by Buch, Kresch, and Tamvakis [BKTb, BKTa]. Their formula expresses an arbitrary Schubert class as a polynomial in the special Schubert classes, it is defined by means of Young's raising operators. We can regard this polynomial expression as a certain "combinatorial interpolation" between the Jacobi-Trudi determinant and the Schur Pfaffian.

This paper is mainly concerned with the equivariant Giambelli problem for the non-maximal isotropic Grassmannians in the symplectic case. In [START_REF] Wilson | Equivariant Giambelli Formulae for Grassmannians[END_REF] , Wilson employed the raising operators to define the double theta polynomials, and proved that these polynomials satisfy the equivariant Chevalley formula for the non-maximal symplectic Grassmannian. In [START_REF] Wilson | Equivariant Giambelli Formulae for Grassmannians[END_REF], it was further conjectured that the double theta polynomials would be equal to the double Schubert polynomials in [START_REF] Ikeda | Double Schubert polynomials for the classical groups[END_REF], a distinguished family of polynomials which are identified with the equivariant Schubert classes of the flag variety in the infinite rank setting.

The main result of this paper provides a formula expressing each double Schubert polynomial associated to the isotropic Grassmannians as a sum of Pfaffians whose entries are Wilson's double theta polynomials corresponding to the special Schubert classes. This immediately leads to a proof of Wilson's conjecture, because the raising operator formula can be rewritten as a Pfaffian sum by a formal computation. Our method to prove the main result is to use the left divided difference operators that are essential in the theory of (double) Schubert polynomials and exist only in its equivariant setup.

This article is a short version of the paper [START_REF] Ikeda | Pfaffian sum formula for symplectic Grassmannians[END_REF]. Detailed proofs are omitted. The only content which is not included in [START_REF] Ikeda | Pfaffian sum formula for symplectic Grassmannians[END_REF] is about a combinatorial description of the Bruhat order on a certain parabolic quotient of the type C Weyl group (Theorem 2.8).

Double Schubert polynomials of type C

We first recall the definition and fundamental properties of the double Schubert polynomials introduced in [START_REF] Ikeda | Double Schubert polynomials for the classical groups[END_REF]. Then we discuss the correspondences among the k-strict partitions, the characteristic indices and the parabolic quotients of the Weyl group.

Double Schubert polynomials

Weyl group for type C Let W ∞ be the infinite hyperoctahedral group which is defined by the generators s i , i = 0, 1, . . ., and the relations

s 2 i = e (i ≥ 0), s i s j = s j s i (|i -j| ≥ 2), s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 , s i s i+1 s i = s i+1 s i s i+1 (i ≥ 1).
(2.1)

We identify W ∞ with the group of all permutations w of Z\{0} such that w(i) = i for only finitely many i ∈ Z\{0}, and w(i) = w( ī) for all i where ī := -i. The generators, often referred to as the simple reflections, are identified with the transpositions s 0 = (1, 1) and s i = (i + 1, i)(i, i + 1) for i ≥ 1. Let W n be the subgroup of W ∞ generated by s 0 , s 1 , . . . , s n-1 . Or equivalently, it consists of elements w ∈ W ∞ such that w(i) = i for all i > n. The one-line notation of an element w ∈ W ∞ is the sequence w = (w(1)w(2)w(3) • • • ). We often write the one-line notation of w ∈ W n by the finite sequence (w(1)w(2)

• • • w(n)). The length of w ∈ W ∞ is denoted by (w).
The ring of Double Schubert Polynomials for type C Let x = (x 1 , x 2 , . . .) be a sequence of indeterminates. Let Q r (x) (r ≥ 0) be the functions defined by the generating function

∞ r=0 Q r (x)u r = ∞ i=1 1 + x i u 1 -x i u . (2.2)
Let Γ be the ring generated by Q r (x) (r ≥ 0). Let R ∞ be the polynomial ring Γ[t, z] in the indeterminates t = (t 1 , t 2 , . . .), and z = (z 1 , z 2 , . . .) with the coefficients in Γ.

There is a distinguished

Z[t]-basis C w (w ∈ W ∞ ) of R ∞ called the double Schubert polynomials of type C.
To characterize this basis, we first define two actions of W ∞ on R ∞ (the right and left actions), and then introduce the right and left divided difference operators.

The right action on R ∞ as a Z[t]-algebra For each i ≥ 1, define the Z[t]-algebra automorphism s z i of R ∞ by s z i (z i ) = z i+1 , s z i (z i+1 ) = z i , s z i (z j ) = z j (j = i, i + 1), and
s z i (Q r (x)) = Q r (x). Also define the Z[t]-algebra automorphism s z 0 of R ∞ by s z 0 (z 1 ) = -z 1 , s z 0 (z i ) = z i (i ≥ 1), and s z 0 Q r (x 1 , x 2 , . . .) = Q r (z 1 , x 1 , x 2 , . . .).
These satisfy the Coxeter relations (2.1), and therefore define an action of W ∞ on R ∞ as a Z[t]-algebra.

The left action on R ∞ as a Z[z]-algebra Consider the involution ω of the ring R ∞ :

ω(t i ) = -z i , ω(z i ) = -t i , ω(Q r (x)) = Q r (x).
For each i ≥ 0, we define the Z[z]-algebra automorphism s t i of R ∞ by s t i := ωs z i ω. In particular, we have

s t 0 Q r (x 1 , x 2 , . . .) = Q r (-t 1 , x 1 , x 2 , .
. .). These satisfy the Coxeter relations (2.1), and therefore define an action of W ∞ on R ∞ as a Z[z]-algebra.

Divided difference operators

We define the right and left divided difference operators respectively by

∂ i f = f -s z i f ω(α i )
and

δ i f = f -s t i f α i (i ≥ 0, f ∈ R ∞ ),
where α i (i ≥ 0) are the simple roots defined by α 0 = 2t 1 and α i = t i+1 -t i for all i ≥ 1.

Theorem 2.1 ([IMN11]

) There exists a unique

Z[t]-free basis {C w (z, t; x) | w ∈ W ∞ } of R ∞ satisfying the equations ∂ i C w = C wsi if (ws i ) < (w) 0 otherwise and δ i C w = C siw if (s i w) < (w) 0 otherwise , (2.3)
for all i ≥ 0, and such that C w has no constant term except C e = 1.

The k-Grassmannian elements and the invariant subring

R (k) ∞ Let W (k) be the subgroup of W ∞ generated by all s i , i = k. Let W (k)
∞ be the set of minimum length coset representatives for W ∞ /W (k) , and it is given by

W (k) ∞ = {w ∈ W ∞ | (ws i ) > (w) for all i = k}. We denote W (k) ∞ ∩ W n by W (k) n . An element of W (k)
∞ is called k-Grassmannian and it is given by the following one-line notation:

w = (v 1 • • • v k |ζ 1 • • • ζ s u 1 u 2 • • • ); 0 < v 1 < • • • < v k , ζ 1 < • • • < ζ s < 0 < u 1 < u 2 < • • • .
(2.4)

We insert a vertical line after v k to indicate that w is regarded as a k-Grassmannian element. For example,

(13| 4256 • • • ) is a 2-Grassmannian element in W ∞ . Lemma 2.2 Let w ∈ W (k)
∞ . We have (s 0 w) < (w) if and only if

(L0) w = (• • • | • • • 1 • • • ), i.e., ζ s = 1.
Suppose i ≥ 1. Then (s i w) < (w) if and only if one of the following holds:

(L1) w = (• • • | • • • i + 1 • • • i • • • ), i.e.
, ζ p = i + 1 and u q = i for some p and q;

(L2) w = (• • • i • • • | • • • i + 1 • • • ), i.e.
, ζ p = i + 1 and v q = i for some p and q;

(L3) w = (• • • i + 1 • • • | • • • i • • • ), i.e.
, u p = i and v q = i + 1 for some p and q.

In particular, for any i ∈ {0, 1, 2, . . . , }, (s i w) = (w) -1 implies

s i w ∈ W (k) ∞ . Let R (k) ∞ be the Z[t]-subalgebra of R ∞ fixed by the right action of W (k) : R (k) ∞ := {f ∈ R ∞ | s z i (f ) = f (∀i = k)}.
We can show that R

(k) ∞ = w∈W (k) ∞ Z[t]C w . Furthermore, the double Schubert polynomials C w for the k-Grassmannian elements w ∈ W (k)
∞ are characterized in terms of the left divided difference operators as follows. For f ∈ R ∞ , let f | ∅ denote the polynomial in Z[t] obtained from f by the substitutions x i → 0 and z i → t i for all i.

Proposition 2.3 If a family F w , w ∈ W (k) ∞ of elements of R (k)
∞ satisfies the following conditions

δ i F w = F siw if (s i w) < (w) 0 if (s i w) > (w) ;
(2.5)

F w | ∅ = δ w,e , (2.6
)

then F w = C w for all w ∈ W (k)
∞ .

The k-strict partitions

A partition λ is a sequence of non-increasing non-negative integers (λ 1 , λ 2 , . . . ) where only finitely many

λ i 's are positive. It is called k-strict if no part greater than k is repeated. Let P (k)
∞ be the set of all kstrict partitions. Let P (k)

n ⊂ P (k)
∞ be the subset of all k-strict partitions whose Young diagrams fit in the (n -k) × (n + k) rectangle. We often denote λ by a finite sequence (λ 1 , . . . , λ r ), assuming λ i = 0 for all i > r, and also denote the sum of the positive entries by |λ|.

Lemma 2.4 ([BKTb]

) There is a bijection

P (k) ∞ ∼ = W (k) ∞ (λ ↔ w (k) λ ) such that |λ| = (w (k) λ ).
We describe the bijection below. Let λ = (λ 1 , λ 2 , . . . ) ∈ P (k) ∞ . Let s be an index such that λ s > k and λ s+1 ≤ k. Let ζ i := λ i -k for i ∈ {1, . . . , s}. Let µ = (µ 1 , . . . , µ k ) be the conjugate of the partition (λ s+1 , λ s+2 , . . . ). For i ∈ {1, . . . , k}, define

v i = s + µ k+1-i + i -{p | ζ p > s + µ k+1-i + i}.
The signed permutation w λ is given by the one-line notation (2.4) where u 1 , u 2 , . . . are the increasing positive integers determined uniquely by v and ζ.

Conversely, let w ∈ W (k)

∞ with the one-line notation (2.4). Define a partition ν = (ν 1 , ν 2 , . . . ) by (1) µ = (1 324) = s 2 s 1 s 0 s 1 respectively. We see that w

ν i = {p | v p > u i }. Then a k-strict partition λ defined by setting λ i = ζ i + k if 1 ≤ i ≤ s and λ i = ν i-s if s + 1 ≤ i.

Characteristic index and Bruhat order on W (k) ∞

The following indexing set for W (k) ∞ turns out to be useful for both the understanding of Bruhat order and the computation in the left divided difference operators.

Definition 2.6 (Characteristic index) For each w ∈ W (k)
∞ with the one-line notation (2.4), we define the associated characteristic index by

χ = (χ 1 , χ 2 , . . . ) = (ζ 1 -1, ζ 2 -1, . . . , ζ s -1, -u 1 , -u 2 , . . . ). In particular, if w ∈ W (k) n , then χ i = -i -k for all i > n -k.
If χ and λ are the characteristic index and the k-strict partition associated to w respectively, then we have

χ j = λ j -k -j + {i | i < j, λ i + λ j > 2k + j -i}.
For a strictly decreasing sequence χ = (χ 1 , χ 2 , . . . ) of integers, let

b χ i = χ i + 1 if χ i ≥ 0 -χ i if χ i < 0, for all i = 1, 2, . . . . Definition 2.7 Let Ξ (k)
∞ be the set of all strictly decreasing sequences χ = (χ 1 , χ 2 , . . . ) of integers such that b χ i 's are all distinct, {1, 2, . . . }\{b χ i , i = 1, 2, . . . } is a finite set of cardinality k, and

χ i = -i -k for sufficiently large i. Let Ξ (k) n be the subset of Ξ (k) ∞ consisting of χ such that χ i = -i -k for all i > n -k. Define a partial order on Ξ (k) ∞ by χ ≤ χ if and only if χ i ≤ χ i all i = 1, 2, . . . . Theorem 2.8 The map W (k) ∞ → Ξ (k)
∞ associating the characteristic indices is an order preserving bijection. Furthermore this map also induces an order preserving bijection

W (k) n ∼ = Ξ (k) n .
The proof of this theorem is postponed until Section 3.3.

Example 2.9 Let λ and µ be the partitions given in Remark 2.5. The corresponding characteristic indices are χ λ = (0, -2, -4) and χ µ = (2, -2, -4) respectively. We find χ λ ≤ χ µ . This verifies Theorem 2.8 since we know w λ ≤ w µ from Remark 2.5.

Equivariant Schubert classes of symplectic Grassmannians

We explain the fact that the double Schubert polynomials represent the equivairant Schubert classes.

Flag varieties of type C

Let G = Sp 2n (C) be the complex symplectic group of rank n. Let B be a Borel subgroup of G, and T the maximal torus contained in B (see [START_REF] Ikeda | Double Schubert polynomials for the classical groups[END_REF] for the precise conventions). The flag variety Fl n is defined as the quotient space G/B. The Weyl group N G (T )/T is identified with W n . For each w ∈ W n , there is the point e w in Fl n . Let B -be the Borel subgroup such that B ∩ B -= T . The Schubert variety X w is defined as the Zariski closure of B --orbit of the point e w ∈ Fl n . The codimension of X w is precisely the length (w) of w. Since X w is a T -stable subvariety of Fl n it defines the fundamental class [X w ] T in the integral T -equivariant cohomology ring H * T (Fl n ). We call [X w ] T the equivariant Schubert class associated to w ∈ W n . We regard H * T (Fl n ) as a Z[t]-algebra by identifying H * T (pt) = Z[t 1 , . . . , t n ] and defining the action of t i to be trivial for all i > n.

Theorem 3.1 ([IMN11]) There is a canonical Z[t]-algebra homomorphism π n : R ∞ -→ H * T (Fl n ) such that π n sends C w (z, t; x) to [X w ] T if w ∈ W n and to zero if w ∈ W n .

Symplectic Grassmannian and its Schubert varieties

We fix a non-negative integer k. For any positive integer n ≥ k, let SG k n denote the Grassmannian of (n -k)-dimensional isotropic subspaces in C 2n equipped with a symplectic form. There is a maximal parabolic subgroup P k of the symplectic group G = Sp 2n (C) such that SG k n can be realized as the homogeneous space G/P k .

Let e 1 , . . . , e n , e * 1 , . . . , e * n be a standard symplectic basis of C 2n , i.e., define the symplectic form by e i , e j = e * i , e * j = 0, e i , e * j = δ ij . For 1 ≤ i ≤ n, define a complete flag

F • : F n ⊂ • • • ⊂ F 1 ⊂ F 1 ⊂ • • • ⊂ F n (3.1)
by F i = e i , . . . , e n and F i = e * i , . . . , e * 1 + F 1 . For each λ ∈ P (k)

n , the corresponding Schubert variety Ω λ with respect to F • can be defined as

Ω λ := {V ∈ SG k n | dim(V ∩ F w (k) λ (k+j) ) ≥ j (1 ≤ j ≤ n -k)}. (3.2)
In particular, the special Schubert varieties are the ones associated to one-line k-strict partitions:

Ω r = {V ∈ SG k n | dim(V ∩ F r-k-1 ) ≥ 1}, for 1 ≤ r ≤ n + k. Their classes [Ω r ]
T are called the special Schubert classes. They are equal to the r-th equivariant Chern classes c r (Q) of the universal quotient bundle

Q over SG k n . For 1 ≤ j ≤ n -k, we have dim F w (k) λ (k+j) = n -χ j . (3.3)
This explains how naturally the characteristic index is defined. There is the following commutative diagram :

R (k) ∞ π (k) n / / R ∞ πn H * T (SG k n ) p * / / H * T (Fl n )
where the horizontal arrow p * in the second row is the pullback of the natural projection p :

Fl n → SG k n and π (k) n is obtained by restricting π n to R (k) ∞ . If λ ∈ P (k) n , we have p * [Ω λ ] T = [X w (k) λ ] T and π (k) n (C w (k) λ (z, t; x)) = [Ω λ ] T .
In particular, the special Schubert class [Ω r ] T of degree r is the image of C w (k) r (z, t; x) where w

(k) r is the element of W (k)
∞ corresponding to the partition with r boxes in one row.

Proof of Theorem 2.8

By the definition of the characteristic index, it is clear that the map is a bijection. Choose n such that w, w ∈ W (k) n . Let χ and χ be the characteristic indices of w and w respectively. Let λ and λ be the strict partitions corresponding to w and w respectively. We reindex the flag F • defined in (3.1), and consider the complete flag U

• : 0 = U n ⊂ • • • U n-1 • • • ⊂ U 0 ⊂ U -1 ⊂ • • • ⊂ U -n = C 2n where for each i ∈ {1, 2, . . . , n}, U -i = F ī and U i = F i-1 . Note that dim U i = n -i for all -n ≤ i ≤ n by (3.3). Then the Schubert variety Ω λ is given by Ω λ = {V ∈ SG k n | dim(V ∩ U χi ) ≥ i, i = 1, . . . , n -k}.
It is well-known that Ω λ ⊃ Ω λ if and only if w ≤ w . Therefore w ≤ w if and only if, for all i = 1, . . . , n -k, we have

U χ i ⊂ U χi i.e. χ i ≤ χ i .
4 Pfaffian sum formula for the equivariant Schubert classes

Double theta polynomials

Our goal is to give an explicit closed formula to describe C w (z, t; x) (w ∈ W

(l) r (x, z|t) for l, r ≥ 0 by ∞ r=0 k ϑ (l) r (x, z|t) • u r = ∞ i=1 1 + x i u 1 -x i u k i=1 (1 + z i u) l i=1 (1 -t i u), ∞ r=0 k ϑ (-l) r (x, z|t) • u r = ∞ i=1 1 + x i u 1 -x i u k i=1 (1 + z i u) l i=1 1 1 + t i u .
For r < 0, we set k ϑ (l) r (x, z|t) = 0. We omit k when it is made clear by the context. Under π n , the indeterminates z i correspond to the Chern roots of the tautological bundles and those theta polynomials Although the above definition of the double theta polynomials appears slightly different from the one in Wilson's thesis [START_REF] Wilson | Equivariant Giambelli Formulae for Grassmannians[END_REF], one recovers Wilson's definition after applying appropriate changes of indices. See [START_REF] Ikeda | Pfaffian sum formula for symplectic Grassmannians[END_REF] for details. We can prove the following lemma from the definitions by direct computations.

Multi Schur-Pfaffian

For each i ≥ 1, let c (i) be an infinite sequence of indeterminates c is defined as follows:

• for m = 1 we set Pf[c

r ] = c

(1) r .

• for m = 2, we set Pf[c

(1) r1 c

(2) r2 ] = c

(1) r1 c

(2) r2 + 2 r2 s=1 (-1) s c

(1) r1+s c

(2) r2-s .

• for any odd m ≥ 3, we set Pf[c The following properties follow from the above definition of Pfaffian by the induction on m.

Remark 2. 5

 5 By the bijection in Lemma 2.4, the natural Bruhat order on W (k) ∞ does not correspond to the order on P (k) ∞ defined by inclusions of Young diagrams. For example, consider 1-strict partitions λ = (21) and µ = (4). The corresponding permutations are w (1) λ = (3 124) = s 2 s 0 s 1 and w

µ

  , while λ ⊂ µ.

∞

  ) as a polynomial in terms of the double Schubert polynomials C wr (z, t; x) corresponding to the special classes [Ω r ] T . Definition 4.1 Define k ϑ

  r (x, z|t) map to Chern classes of certain virtual bundles. In particular, Wilson in [Wil10, Prop. 6] proved C wr (z, t; x) = k ϑ (r-k-1) r (x, z|t). (4.1)

Lemma 4. 2

 2 (a) For all i ≥ 0, we have δ i ϑ(l) r = 0 if l = ±i and δ i ϑ if l = ±i.(b) For i = 0, we have δ i (ϑ

r

  , r ∈ Z. Consider the polynomial ring Z[c] := Z[c (i) r , i ≥ 1, r ∈ Z]. The multi Schur-Pfaffian Pf[c (1) r1 . . . c (m) rm ] ∈ Z[c]

•

  for any even m ≥ 4, we set Pf[c

Acknowledgements

We would like to thank Dave Anderson, Anders Skovsted Buch, Andrew Kresch, Changzheng Li, Leonardo Mihalcea, and Masaki Nakagawa, Hiroshi Naruse, Harry Tamvakis for helpful conversations and for their comments.

Pfaffian sum formula (Main Theorem)

Definition 4.4 For all (r 1 , . . . , r m ), (l 1 , . . . , l m ) ∈ Z m , let

rm c=ϑ (l) where | c=ϑ (l) means that we substitute ϑ (li) s for c

s for all i ∈ {1, . . . , m} and s ∈ Z.

The main result of this paper is the following.

Theorem 4.5 (Main Theorem) Let λ be a k-strict partition in P (k) n . We have

where I runs over all subsets of D(λ) and

Once we read the formula in terms of raising operators, the following corollary is immediate.

In the case when λ is contained in the (n -k) × k rectangle, the result was proved by Wilson [START_REF] Wilson | Equivariant Giambelli Formulae for Grassmannians[END_REF].

Related results

Anderson and Fulton [AF] defined a notion of vexillary signed permutation in type B,C, and D. They showed the double Schubert polynomials associated to vexillary signed permutations are given by explicit Pfaffian formulas. Naruse [Nar] also independently proved a formula that expresses the corresponding double Schubert polynomials as a specialization of the factorial Qand P -functions. Since our formula also expresses some Schubert classes as single Pfaffians, there is an overlap between our results and the results of [AF]. However, not all k-Grassmannian permutations are vexillary and there are non-vexillary k-Grassmannian permutations whose corresponding classes are written as single Pfaffians, e.g. 13| 54 2 is not vexillary but C 13| 54 2 is a single Pfaffian as above. The proof of our main theorem is essentially based on the following two propositions. We can show them by using the inductive definition of the multi Schur-Pfaffian together with Lemma 4.2.

Proof of Main Theorem

Proposition 5.1 Suppose l ≥ 0 and r > k + l. Then Pf[ϑ

(b) Let i ≥ 0. Suppose that l p ∈ {±i} for some p ∈ {1, . . . , m} and that l q ∈ {±i} for all q = p.

(c) Let i > 0. Suppose that l p = i and l q = -i for some p < q and that l s = ±i for all s ∈ {p, q}.

Then we have

Key lemma

For w ∈ W n . We proceed by the induction on (w ∨ ). The claim obviously holds if

n satisfying (λ ) = (λ)+1. By the induction hypothesis we have

where χ = (χ 1 , . . . , χ n-k ) is the characteristic index of λ . Lemma 2.2 gives the 4 possible cases of the characteristic index χ in terms of χ. In each case, we apply δ i to the right hand side of (5.1) by using Proposition 5.2 and obtain Θ

(n,k) w

. Since δ w ∨ = δ i δ w ∨ λ , the desired formula holds.

Proof of Main Theorem

First note that the polynomial defined by Θ

(n,k) w is independent of n, i.e., for all m ≥ n, and w ∈ W

. We can see this fact from the definition of the multi-Schur Pfaffian. Thus, for each w ∈ W By observing that (s i w) = (w) -1 implies δ i δ w ∨ = δ (siw) ∨ , the first half of the condition (2.5) follows from Lemma 5.3. For the second half of the condition (2.5), first we see that δ j Θ w (n,k) max = 0 for all j = k from Proposition 5.1 and Proposition 5.2 (a). Now the claim follows from

∞ , then there exists j = k such that δ i δ w ∨ = δ w ∨ δ j .

It remains to show that the condition (2.6) holds. It is obvious that Θ e | e = 1. For w = 0, we expand Θ w as a polynomial in terms of the ϑ-functions ϑ (l) r by using the definition of Pfaffian. Note that each monomial contains ϑ (χ1) λ1+a I 1 +j as its first factor. We finish the proof by showing the substitution makes this factor vanish by using the following easy observation: ϑ

An alternative proof of Main Theorem

Let w n,0 be the longest element in W n . In [IMN11, Theorem 1.2], the authors proved that C wn,0 can be obtained from the (factorial) Q-function by a certain specialization of variables. In terms of Pfaffian and the theta polynomials k ϑ l r := k ϑ l r (x, z|t), we have


Takuro Abe
email: abe.takuro.4c@kyoto-u.ac.jp.

Daisuke Suyama
email: dsuyama@math.sci.hokudai.ac.jp.

Shuhei Tsujie
email: tsujie@math.sci.hokudai.ac.jp.

The freeness of Ish arrangements

Keywords: Hyperplane arrangement, Ish arrangement, Shi arrangement, Coxeter arrangement, Supersolvable arrangement

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.

Introduction

Let K be a field of characteristic 0 and {x 1 , . . . , x } a basis for the dual space (K ) * of the -dimensional vector space K . The Coxeter arrangement Cox( ) of type A -1 (also called the braid arrangement) is Cox( ) := {{x i -

x j = 0} | 1 ≤ i < j ≤ } ,
where {x = k} (x ∈ (K ) * , k ∈ K) is the affine hyperplane {v ∈ K | x(v) = k}. Then the Shi arrangement Shi( ) and the Ish arrangement Ish( ) are defined by

Shi( ) := Cox( ) ∪ {{x i -x j = 1} | 1 ≤ i < j ≤ } , Ish( ) := Cox( ) ∪ {{x 1 -x j = i} | 1 ≤ i < j ≤ } .
The Shi arrangement originally defined over R was introduced by J.Y. [START_REF] Shi | The Kazhdan-Lusztig cells in certain affine Weyl groups[END_REF]] in the study of the Kazhdan-Lusztig representation theory of the affine Weyl groups. The Ish arrangement also originally defined over R was introduced by Armstrong in [START_REF] Armstrong | Hyperplane arrangements and diagonal harmonics[END_REF]]. He gave a new interpretation of the q, t-Catalan numbers of Garsia and Haiman by using these two arrangements. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement in [START_REF] Armstrong | Hyperplane arrangements and diagonal harmonics[END_REF]; [START_REF] Armstrong | The Shi arrangement and the Ish arrangement[END_REF]].

Let A be an arrangement in K . Let L(A) be the set of nonempty intersections of hyperplanes in A, which is partially ordered by the reverse inclusion of subspaces. Define the Möbius function µ : L(A) → Z as follows:

µ(K ) = 1, µ(X) = - K ≤Y <X µ(Y ) (X = K ).
Then the characteristic polynomial χ(A, t) ∈ Z[t] of A is defined by

χ(A, t) = X∈L(A) µ(X)t dim X .
The following theorem is one of the similarities pointed out by Armstrong.

Theorem 1.1 [START_REF] Armstrong | Hyperplane arrangements and diagonal harmonics[END_REF]; [START_REF] Headley | On a family of hyperplane arrangements related to the affine Weyl groups[END_REF]]) The characteristic polynomial of the Shi arrangement and the Ish arrangement are given by

χ(Shi( ), t) = χ(Ish( ), t) = t(t -) -1 .
Let {x 1 , . . . , x , z} be a basis for V * of V := K +1 . Then, as in [ [START_REF] Orlik | of Grundlehren der Mathematischen Wissenschaften[END_REF], Definition 1.15)], we have the cone c(Ish( )) over the Ish arrangement which is a central arrangement (Namely, an arrangement whose hyperplanes pass through the origin) in V defined by

Q (c(Ish( ))) = z 1≤i<j≤ (x i -x j )(x 1 -x j -iz) = 0.
Let S be the symmetric algebra of the dual space V * . S can be identified with the polynomial ring K[x 1 , . . . , x , z]. Let Der(S) be the module of derivations of S

Der(S) := {θ : S → S | θ is K-linear, θ(f g) = f θ(g) + θ(f )g for any f, g ∈ S}.
Then, for a central arrangement A in V , the module of logarithmic derivations D(A) of A is defined to be

D(A) := {θ ∈ Der(S) | θ(Q(A)) ∈ Q(A)S} = {θ ∈ Der(S) | θ(α H ) ∈ α H S for any H ∈ A},
where Q(A) is the defining polynomial of A and α H is a linear form such that ker(α H ) = H. We say that A is free if D(A) is a free S-module. Then D(A) has a homogeneous basis {θ 0 , . . . , θ } and the tuple of degrees exp A = (deg θ 0 , . . . , deg θ ) is called the exponents of A.

The main purpose of this paper is to settle a problem of whether the Ish arrangements are free or not, which was posed by [START_REF] Armstrong | Hyperplane arrangements and diagonal harmonics[END_REF][START_REF] Armstrong | The Shi arrangement and the Ish arrangement[END_REF]Rhoades, 2012, p. 1527, (3))]. We define a new class of arrangements which is a generalization of the Ish arrangements and will characterize free arrangements in this class.

Definition 1.2 Let N = (N 2 , N 3 , . . . , N ) be a tuple of finite subsets N j in K. Define the N -Ish arrangement Ish(N ) by

Ish(N ) := {{x 1 -x j = a} | 2 ≤ j ≤ , a ∈ N j } ∪ {{x i -x j = 0} | 2 ≤ i < j ≤ } .
We say that N is a nest if there exists a permutation w of {2, . . . , } such that

N w(2) ⊆ N w(3) ⊆ • • • ⊆ N w( ) .
In particular, when N j = {0, 1, . . . , j -1} for each j, the N -Ish arrangement Ish(N ) is the Ish arrangement Ish( ). We denote the cone over the N -Ish arrangement c(Ish(N )) by I = I N . The defining polynomial of I can be expressed as

Q(I) = z   j=2 a∈Nj (x 1 -x j -az)     2≤i<j≤ (x i -x j )   .
Our main results are as follows:

Theorem 1.3 The following four conditions are equivalent:

(1) N is a nest.

(2) I N is supersolvable.

(3) I N is inductively free.

(4) I N is free.

The definitions of supersolvable and inductively free arrangements will be mentioned in Section 2. Note that the implications (2) ⇒ (3) ⇒ (4) are general properties for arrangements [START_REF] Orlik | of Grundlehren der Mathematischen Wissenschaften[END_REF]]. This theorem asserts that there are no differences among these properties for N -Ish arrangements.

Theorem 1.4 Let N = (N 2 , N 3 , . . . , N j ) with N 2 ⊆ N 3 ⊆ • • • ⊆ N j . Define homogeneous derivations θ 0 , θ 1 , . . . , θ by θ 0 := i=1 ∂ ∂x i , θ 1 = i=1 x i ∂ ∂x i + z ∂ ∂z , θ k := k s=2 a∈N k (x 1 -x s -az) t=k+1 (x s -x t ) ∂ ∂x s (2 ≤ k ≤ ).
Then θ 0 , θ 1 , . . . , θ form a basis for D(I N ). In particular, the exponents are given by

exp I N = (0, 1, |N 2 | + -2, |N 3 | + -3, . . . , |N |),
where |N j | denotes the cardinality of N j .

Corollary 1.5 The cone over the Ish arrangement c(Ish( )) is free with exponents exp(c(Ish( ))) = (0, 1, , , . . . , ( -1) times

). Moreover the homogeneous derivations

θ 0 = i=1 ∂ ∂x i , θ 1 = i=1 x i ∂ ∂x i + z ∂ ∂z , θ k = k s=2 k-1 i=0 (x 1 -x s -iz) t=k+1 (x s -x t ) ∂ ∂x s (2 ≤ k ≤ )
form a basis for D(c(Ish( ))).

If an arrangement A is a free arrangement, then the characteristic polynomial of A can be expressed by using its exponents: [START_REF] Terao | Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula[END_REF]]) If an arrangement A is free with exponents (d 1 , . . . , d ), then the characteristic polynomial of A splits as

Theorem 1.6 ([
χ(A, t) = i=1 (t -d i ).
Since we have the relation between the characteristic polynomials of A and cA

χ(cA, t) = (t -1)χ(A, t),
we obtain a new proof of Theorem 1.1 from Corollary 1.5 and Theorem 1.6.

The complement M (A) := K \ ∪ H∈A H of a supersolvable arrangement A has very interesting properties: If K = C, the complement M (A) is fiber type [START_REF] Terao | Modular elements of lattices and topological fibration[END_REF]]. In particular, M (A) is a K(π, 1) space, i.e., the homotopy groups π i (M (A)) = 0 for i ≥ 2. When K = R, the complement M (A) is a disjoint union of chambers. For chambers C, C , define d(C, C ) by the number of hyperplanes in A separating C from C . Björner, Edelman, and Ziegler [START_REF] Björner | Hyperplane arrangements with a lattice of regions[END_REF]] gave the wall-crossing formula as follows: There exists a base chamber B of A such that

C∈Ch(A) t d(B,C) = i=1 (1 + t + • • • + t di ),
where (d 1 , . . . , d ) is the exponents of A and Ch(A) denotes the set of all chambers of A. Therefore, we derive the following corollary from our main theorems 1.3 and 1.4.

Corollary 1.7 Let N = (N 2 , N 3 , . . . , N j ) with N 2 ⊆ N 3 ⊆ • • • ⊆ N j .
(1) If K = C, then the complement M (I N ) of the cone over the N -Ish arrangement I N is K(π, 1).

(2) If K = R, then there exists a base chamber B ∈ Ch(I N ) such that

C∈Ch(I N ) t d(B,C) = t i=2 (1 + t + • • • + t |Ni|+ -i ).
The organization of this paper is as follows. In Section 2, we review the theory of supersolvable arrangements and prove Theorem 1.3. In Section 3, we verify Theorem 1.4 applying Saito's criterion. In Section 4, we recall the deleted arrangement Shi(G) and Ish(G) defined by Armstrong and Rhoades in [START_REF] Armstrong | The Shi arrangement and the Ish arrangement[END_REF]] and prove that Shi(G) and Ish(G) share the freeness.

2 Supersolvability and freeness of I For an arrangement A, let L(A) be the set of nonempty intersections of hyperplanes in A. If A is central, then L(A) is a geometric lattice with the order by reverse inclusion: X ≤ Y ⇔ Y ⊆ X. In the rest of this section, "arrangement" means "central arrangement". The rank of an arrangement A, denoted by rank(A), is the codimension of ∩ H∈A H. We say that A is essential if rank(A) is equal to the dimension of the ambient space of A.

An arrangement A is supersolvable if the intersection lattice L(A) is supersolvable as defined by Stanley [START_REF] Stanley | Supersolvable lattices[END_REF]]. The following lemma is widely known.

Lemma 2.1 [START_REF] Terao | Modular elements of lattices and topological fibration[END_REF]]) An arrangement A is supersolvable if and only if there exists a filtration

A = A ⊇ A -1 ⊇ • • • ⊇ A 1 such that (1) rank(A i ) = i (i = 1, 2, . . . , ).
(2) For any H, H ∈ A i with H = H , there exists some

H ∈ A i-1 such that H ∩ H ⊆ H .
Let A be an arrangement. For a hyperplane H ∈ A, define arrangements

A := A \ {H} and A := {H ∩ H | H ∈ A } .
The tuple (A, A , A ) is called the triple of arrangements with respect to H. For a triple (A, A , A ), the Addition Theorem [Terao (1980a,b)] asserts that if A and A are free and exp A ⊂ exp A , then A is free.

Definition 2.2 Define the inductive freeness by the following:

(1) The empty arrangement is inductively free.

(2) A is inductively free if there exists H ∈ A such that A and A are inductively free and exp A ⊂ exp A .

Thanks to the Addition Theorem, the inductive freeness implies the freeness. Moreover, it is also known that the the supersolvability implies the inductive freeness (see [ [START_REF] Orlik | of Grundlehren der Mathematischen Wissenschaften[END_REF], Theorem 4.58)] for example). We will use the following lemma which is a part of the Addition-Deletion Theorem:

Lemma 2.3 ([ [START_REF] Orlik | of Grundlehren der Mathematischen Wissenschaften[END_REF], Theorem 4.46)]) Let (A, A , A ) be a triple. Suppose that A is an essential arrangement of rank 3 and that arrangements A and A are free with exp(A ) = (1, a, b) and exp(A ) = (1, c). If c ∈ {a, b} then A is not free.

We are now prepared to prove Theorem 1.3.

Proof of Theorem 1.3: (1) ⇒ (2) Without loss of generality, we may assume that

N 2 ⊇ N 3 ⊇ • • • ⊇ N .
For each i ∈ {1, 2, . . . , }, define X i ∈ L(I) by

X i := {z = x 1 -x 2 = • • • = x 1 -x i = 0} .
Then the rank of the localization I i := I Xi = {H ∈ I | H ⊇ X} is equal to i and we have

I i = {{x 1 -x j = az} | 1 < j ≤ i and a ∈ N j } ∪ {{x j -x k = 0} | 2 ≤ j < k ≤ i} ∪ {{z = 0}}
Hence there exists a filtration

I = I ⊇ I -1 ⊇ • • • ⊇ I 1 .
By Lemma 2.1, we have only to verify that for any H, H ∈ I i with H = H there exists some H ∈ I i-1 such that H ∩ H ⊆ H for each i ∈ {2, . . . , }. We may assume that both H and H do not belong to I i-1 . Then H and H belong to

I i \ I i-1 = {{x 1 -x i = az} | a ∈ N i } ∪ {{x j -x i = 0} | 2 ≤ j < i} .
First, let a and b be distinct elements in

N i . Suppose that H = {x 1 -x i = az} and H = {x 1 -x i = bz}. Then H ∩ H ⊆ {z = 0} ∈ I i-1 . Second, let j and k be distinct integers in {2, . . . , i -1}. Assume that H = {x j -x i = 0} and H = {x k -x i = 0}. Then H ∩ H ⊆ {x j -x k = 0} ∈ I i-1 . Finally, let H = {x 1 -x i = az} and H = {x j -x i = 0} with a ∈ N i and 2 ≤ j < i. Then H ∩ H ⊆ {x 1 -x j = az} ∈ I i-1
by the assumption a ∈ N i ⊆ N j . Thus the cone over the N -Ish arrangement I is supersolvable.

(2) ⇒ (3) ⇒ (4) We have nothing to prove as mentioned before.

(4) ⇒ (1) When = 2, the tuple N = (N 2 ) is a nest. For ≥ 3, we will prove that if N is not a nest then I is not free by induction on . First, let = 3. Then we have N = (N 2 , N 3 ). Let H ∈ I be the hyperplane {x 2 -x 3 = 0} and (I, I , I ) the triple with respect to H. One can verify easily that the homogeneous derivations

3 i=1 x i ∂ ∂x i + z ∂ ∂z , a∈N2 (x 1 -x 2 -az) ∂ ∂x 2 , a∈N3 (x 1 -x 3 -az) ∂ ∂x 3
form a basis for D(I ) (with the non-essential derivation

3 i=1 ∂ ∂xi + ∂ ∂z ). Hence the arrangement I is free with exponents (1, |N 2 |, |N 3 |). The arrangement I is also free with exponents (1, |N 2 ∪ N 3 |) since rank(I ) = 2 and |I | = 1+|N 2 ∪N 3 |. By the assumption, N is not a nest, i.e., N 2 ⊆ N 3 and N 2 ⊇ N 3 , hence we have that |N 2 ∪ N 3 | is strictly larger than both of |N 2 | and |N 3 |.
Therefore, by Lemma 2.3, we have concluded that I is not free. Now suppose that > 3. Since N is not a nest, there exist integers i, j such that N i ⊆ N j and N i ⊇ N j . Define X ∈ L(I) by

X := {z = x 1 -x i = x 1 -x j = 0} .
Then we have

I X = {{x 1 -x k = az} | k ∈ {i, j} and a ∈ N k } ∪ {{x i -x j = 0}, {z = 0}} .
Hence I X is equivalent to c(Ish(N i , N j )) discussed in the above paragraph. Therefore the localization I X is not free, neither is I. 2

3 A basis for D(I)

In this section, we will prove Theorem 1.4. First, we verify that θ 0 , θ 1 , . . . , θ belong to D(I).

Lemma 3.1 Let N = (N 2 , N 3 , . . . , N j ) with N 2 ⊆ N 3 ⊆ • • • ⊆ N j . Then θ 0 = i=1 ∂ ∂x i , θ 1 = i=1 x i ∂ ∂x i + z ∂ ∂z , θ k = k s=2 a∈N k (x 1 -x s -az) t=k+1 (x s -x t ) ∂ ∂x s (2 ≤ k ≤ )
belong to D(I).

Proof: Since θ 0 (α H ) = 0 for any H ∈ I, it belongs to D(I). The Euler derivation θ 1 belongs to D(A) for any central arrangement A, thus θ 1 ∈ D(I). We will show that

θ k ∈ D(I) for 2 ≤ k ≤ . It is obvious that θ k (z) = 0 ∈ zS. Let 2 ≤ i < j ≤ . Case 1. If i < j ≤ k, then θ k (x i -x j ) = a∈N k (x 1 -x i -az) t=k+1 (x i -x t ) - a∈N k (x 1 -x j -az) t=k+1 (x j -x t ) ≡ a∈N k (x 1 -x i -az) t=k+1 (x i -x t ) - a∈N k (x 1 -x i -az) t=k+1 (x i -x t ) (mod x i -x j ) = 0, thus θ k (x i -x j ) ∈ (x i -x j )S. Case 2. If i ≤ k < j, then θ k (x i -x j ) = a∈N k (x 1 -x i -az) t=k+1 (x i -x t ) ∈ (x i -x j )S.
Case 3. If k < i < j, then

θ k (x i -x j ) = 0 ∈ (x i -x j )S. Hence θ k (x i -x j ) ∈ (x i -x j )S for 2 ≤ i < j ≤ . Let 2 ≤ j ≤ and b ∈ N j . Case 1. If j ≤ k, then b ∈ N j ⊆ N k , thus θ k (x 1 -x j -bz) = a∈N k (x 1 -x j -az) t=k+1 (x j -x t ) ∈ (x 1 -x j -bz)S.
Case 2. If k < j, then

θ k (x 1 -x j -bz) = 0 ∈ (x 1 -x j -bz)S.
Hence θ k (x 1 -x j -bz) ∈ (x 1 -x j -bz)S for 2 ≤ j ≤ and b ∈ N j . Therefore we obtain that θ k ∈ D(I). 2

Proof of Theorem 1.4.

First, note that if s = 1, k ≥ 2 then θ k (x s ) = θ k (x 1 ) = 0, and if 2 ≤ k < s then θ k (x s ) = 0.
Thus the determinant of the coefficient matrix of θ 0 , θ 1 , . . . , θ can be calculated as follows:

θ 0 (x 1 ) θ 1 (x 1 ) • • • θ (x 1 ) . . . . . . • • • . . . θ 0 (x ) θ 1 (x ) • • • θ (x ) θ 0 (z) θ 1 (z) • • • θ (z) = 1 x 1 0 • • • 0 1 x 2 θ 2 (x 2 ) • • • θ (x 2 ) . . . . . . 0 . . . . . . 1 x . . . . . . θ (x ) 0 z 0 • • • 0 . = z 1 0 0 • • • 0 1 θ 2 (x 2 ) θ 3 (x 2 ) • • • θ (x 2 ) 1 θ 3 (x 3 ) • • • θ (x 3 ) . . . . . . . . . 1 0 θ (x ) = z k=2 θ k (x k ) = z k=2 a∈N k (x 1 -x k -az) t=k+1 (x k -x t ) = z k=2 a∈N k (x 1 -x k -az) k=2 t=k+1 (x k -x t ) = z k=2 a∈N k (x 1 -x k -az)   2≤k<t≤ (x k -x t )   = Q(I),
where .

= denotes that they are equal up to a nonzero constant multiple. Combining this calculation and Lemma 3.1, we can apply Saito's criterion [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF]] and see that θ 0 , θ 1 , . . . , θ form a basis for D(I).

Freeness of the deleted Ish arrangements

Let K be the complete graph on vertices. We can regard K as the set of directed edges (i, j) (i < j), namely K = {(i, j) | 1 ≤ i < j ≤ }. For a subgraph G ⊆ K , Armstrong and Rhoades [START_REF] Armstrong | The Shi arrangement and the Ish arrangement[END_REF]] defined the deleted arrangements Shi(G) and Ish(G) and showed that they share many properties. In particular, it was proven that Shi(G) and Ish(G) have the same characteristic polynomials by their explicit expressions. The deleted Shi and Ish arrangements are defined by Shi(G) := Cox( ) ∪ {{x i -x j = 1} | (i, j) ∈ G} ⊆ Shi( ), Ish(G) := Cox( ) ∪ {{x 1 -x j = i} | (i, j) ∈ G} ⊆ Ish( ).

Athanasiadis gave a necessary and sufficient condition for the freeness of c(Shi(G)).

second property of (3) implies (w -1 (i), w -1 (k)) ∈ w -1 G, i.e., (i, k) ∈ G. Hence (i) holds. Similarly, if w -1 (j) > w -1 (k) then (ii) holds.

(4) ⇒ (2) For any j, k ∈ {2, . . . , }, it is clear that (i) holds if and only if N j ⊆ N k and (ii) holds if and only if N k ⊆ N j . Therefore every element in N G is comparable. Thus N G is a nest.

2

Combining Theorem 4.1 and Theorem 4.2, we can prove that the following corollary:

Corollary 4.3 The deleted arrangements Shi(G) and Ish(G) share the freeness.

Acknowledgements

The authors thank the referees for careful reading and many useful comments on this article.

Theorem 4.1 [START_REF] Athanasiadis | On free deformations of the braid arrangement[END_REF] Theorem 4.1) Let G ⊆ K be a subgraph. The cone over the deleted Shi arrangement c(Shi(G)) is free if and only if there exists a permutation w of {1, . . . , } such that w -1 G is contained in K , i.e., (i, j) ∈ w -1 G implies i < j, and has the following property:

In this section, we will prove that the property of G in the Theorem 4.1 is also a necessary and sufficient condition for the freeness of c(Ish(G)) by making use of the terminology of the N -Ish arrangements. The problem of whether the cone of the deleted Ish arrangement c(Ish(G)) is free or not is posed by [START_REF] Armstrong | Hyperplane arrangements and diagonal harmonics[END_REF][START_REF] Armstrong | The Shi arrangement and the Ish arrangement[END_REF]Rhoades, 2012, p. 1517)] together with the problem for c(Ish( )).

For a subgraph G ⊆ K , define a tuple of sets N G = (N 2 , . . . , N ) by

It is easy to show that Ish(N G ) = Ish(G).

Theorem 4.2 Let G ⊆ K be a subgraph. Then the following are equivalent:

(1) c(Ish(G)) is free.

(2) N G is a nest.

(3) G has the property in Theorem 4.1.

(4) For any j, k ∈ {2, . . . , }, either of the following two conditions holds:

(ii) If (i, k) ∈ G then (i, j) ∈ G for any i ≤ min{j, k}.

Proof: (1) ⇔ (2) It is obvious from Theorem 1.3.

(2) ⇒ (3) Assume that N G is a nest. Then there exists a permutation w of {1, . . . , } with w(1) = 1 such that

Now, we will prove that w -1 G ⊆ K i.e., (i, j) ∈ w -1 G implies i < j. For any (i, j) ∈ w -1 G, we have (w(i), w(j)) ∈ G. Hence w(i) ∈ N w(j) . Then N w(j) ⊆ N w(i) since w(i) ∈ N w(i) . Since N G is a nest, we have N w(i) ⊆ N w(j) . Therefore i < j, namely (i, j) ∈ K . Thus we have showed that w -1 G ⊆ K .

Suppose that 1 ≤ i < j < k ≤ . Then we have a chain of implications:

This proves that G satisfies the second condition.

(3) ⇒ (4) Fix elements j, k ∈ {2, . . . , } and assume that w -1 (j) < w -1 (k). For any (i, j) ∈ G, we have that (w -1 (i), w -1 (j)) ∈ w -1 G. Since w -1 G ⊆ K , we have w -1 (i) < w -1 (j). Then the


Masaki Watanabe

Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

Keywords: Schubert polynomials, Schubert functors, Kraśkiewicz-Pragacz modules, Schubert calculus

Introduction

The study of Schubert polynomials is an important and interesting subject in algebraic combinatorics. One of the possible methods for studying Schubert polynomials is through the modules introduced by Kraśkiewicz and Pragacz ([KP87], [START_REF] Kraśkiewicz | Schubert functors and Schubert polynomials[END_REF]). Let b be the Lie algebra of all upper triangular matrices. For each permutation w, Kraśkiewicz and Pragacz defined a representation S w of b such that its character with respect to the subalgebra h of all diagonal matrices is equal to the Schubert polynomial S w .

In [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF], the author investigated characterizations of b-modules having a filtration whose subquotients are Kraśkiewicz-Pragacz modules (or KP filtration for short). The main motivation for this study is the investigation of positivity properties of Schubert polynomials. For example, it is a classical fact, usually proved using the cohomology ring of flag variety, that the product S w S v of two Schubert polynomials is a positive sum of Schubert polynomials. If we show that the tensor product S w ⊗ S v of KP modules has a KP filtration, then it gives another proof to this fact. Another example is the question of positivity for the plethystic composition of a Schur function with a Schubert polynomial: if S w = x α + x β + • • • and s λ is a Schur function, is s λ (x α , x β , . . .) a positive sum of Schubert polynomials? A positive answer and s λ is a Schur function, is s λ (x α , x β , . . .) a positive sum of Schubert polynomials? A positive answer to this question, a natural generalization of the Schur-positivity of the plethysm of two Schur functions, follows if we show the Schur-functor image s λ (S w ) of a KP module has a KP filtration. In [START_REF] Watanabe | Tensor product of Kraśkiewicz and Pragacz's modules[END_REF] we gave positive answers to both of these questions.

This extended abstract is a summary of the author's two papers [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF] and [START_REF] Watanabe | Tensor product of Kraśkiewicz and Pragacz's modules[END_REF]: the detailed arguments are given in these two papers and either omitted or only outlined here. The following sections are organized as follows. In Sections 2 and 3 we prepare some basic notations, and introduce the notion of Kraśkiewicz-Pragacz modules. In Section 4 we present some results from [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF] giving a characterization of modules having a filtration by KP modules. In Section 5 we show some applications of our characterization: we present the result that the tensor product of KP modules has a KP filtration, and show that the plethystic composition of a Schur function with a Schubert polynomial is Schubert-positive.

Preliminaries

Let N be the set of all positive integers and let Z ≥0 be the set of all nonnegative integers. A permutation w is a bijection from N to itself which fixes all but finitely many points. Let S ∞ denote the group of all permutations. Let n be a positive integer. Let

S n = {w ∈ S ∞ : w(i) = i (i > n)} and S (n) ∞ = {w ∈ S ∞ : w(n + 1) < w(n + 2) < • • • }.
Sometimes a permutation will be written in one-line form: i.e. if w ∈ S n , we may write [w(1) w(2) • • • w(n)] to mean w. For i < j, let t ij denote the permutation which exchanges i and j and fixes all other points. Let s i = t i,i+1 . For a permutation w, let ℓ(w) = #{i < j : w(i) > w(j)}. Let w 0 ∈ S n be the longest element of S n , i.e.

w 0 (i) = n + 1 -i (1 ≤ i ≤ n). For w ∈ S (n) ∞ we define code(w) = (code(w) 1 , . . . , code(w) n ) ∈ Z n ≥0
by code(w) i = #{j : i < j, w(i) > w(j)}. We define the Rothe diagram D(w) of a permutation w as D(w) = {(i, w(j)) : i < j, w(i) > w(j)}. For a polynomial f = f (x 1 , . . . , x n ) and

1 ≤ i ≤ n -1, we define ∂ i f = f -sif xi-xi+1 . For w ∈ S (n)
∞ we can assign its Schubert polynomial S w ∈ Z[x 1 , . . . , x n ], which is recursively defined by

• S w = x w(1)-1 1 x w(2)-1 2 • • • x w(n)-1 n if w(1) > w(2) > • • • > w(n), and
• S wsi = ∂ i S w if ℓ(ws i ) < ℓ(w).

It is known that the set {S

w : w ∈ S (n) ∞ } constitutes a Z-base for Z[x 1 , . . . , x n ] ([Mac91, (4.13)]).
Let K be a field of characteristic zero. Let b = b n be the Lie algebra of all upper triangular Kmatrices. and let h ⊂ b and n + ⊂ b be the subalgebra of all diagonal matrices and the subalgebra of all For a polynomial f = f (x 1 , . . . , x n ) and

1 ≤ i ≤ n -1, we define ∂ i f = f -sif xi-xi+1 . For w ∈ S (n)
∞ we can assign its Schubert polynomial S w ∈ Z[x 1 , . . . , x n ], which is recursively defined by

• S w = x w(1)-1 1 x w(2)-1 2 • • • x w(n)-1 n if w(1) > w(2) > • • • > w(n), and
• S wsi = ∂ i S w if (ws i ) < (w).

It is known that the set {S

w : w ∈ S (n) ∞ } constitutes a Z-base for Z[x 1 , . . . , x n ] ([Mac91, (4.13)]).
Let K be a field of characteristic zero. Let b = b n be the Lie algebra of all upper triangular Kmatrices. and let h ⊂ b and n + ⊂ b be the subalgebra of all diagonal matrices and the subalgebra of all strictly upper triangular matrices respectively. Let U(b) and U(n + ) be the universal enveloping algebras of b and n + respectively. For a U(b)-module M and λ

= (λ 1 , . . . , λ n ) ∈ Z n , let M λ = {m ∈ M : hm = λ, h m (∀h ∈ h)} where λ, h = λ i h i . If M λ = 0 then λ is said to be a weight of M . If M = λ∈Z n M λ and each M λ has a finite dimension, then M is said to be a weight b-module and we define ch(M ) = λ dim M λ x λ where x λ = x λ1 1 • • • x λn n .
From here we only consider weight b-modules, and write Ext i to mean Ext functors in the category of all weight b-modules. For 1 ≤ i < j ≤ n, let e ij ∈ b be the matrix with 1 at the (i, j)-position and all other coordinates 0. For λ ∈ Z n , let K λ denote the one-dimensional U(b)-module where h ∈ h acts by λ, h and e ij acts by 0. We define ρ = (n -1, n -2, . . . , 0) ∈ Z n , 1 = (1, . . . , 1) ∈ Z n and j = (0, . . . , 1, . . . , 0) ∈ Z n (1 at j-th position).

Kraśkiewicz-Pragacz modules

In [START_REF] Kraśkiewicz | Foncteurs de Schubert[END_REF], Kraśkiewicz and Pragacz defined certain U(b)-modules which we call here Kraśkiewicz-Pragacz modules or KP modules. Here we use the following definition. Let w ∈ S (n)

∞ . Let K n = 1≤i≤n Ku i be the vector representation of b. For each p ∈ N, let {i : (i, p) ∈ D(w)} = {i 1 , . . . , i lp } (i 1 < • • • < i lp ), and let u (p) w = u i1 ∧ • • • ∧ u i lp . Let u w = u (1) w ⊗ u (2) w ⊗ • • • ∈ l1 K n ⊗ l2 K n ⊗ • • • .
Then the KP module S w associated to w is defined as S w = U(b)u w . This module satisfies the following:

Theorem 3.1 ([KP04, Remark 1.6 and Theorem 4.1]) S w is a weight module and ch(S w ) = S w .

Example 3.2 If w = s i , then D(s i ) = {(i, i)}, u si = u i and S si = 1≤j≤i Ku j =: K i . So ch(S si ) x 1 + • • • + x i = S si . Example 3.3 More generally, if w ∈ S (n)
∞ is a Grassmannian permutation with descent at i-th position, i.e. w(1) < • • • < w(i) and w(i + 1) < w(i + 2) < • • • , then the corresponding Rothe diagram is, after a rearrangement of the columns, a Young diagram in French notation (see Figure 2). In this case u w is a lowest-weight vector in a irreducible representation of gl i with lowest weight code(w), and thus S w is isomorphic to this module (seen as a module over b n through the morphism b n e pq → e pq (q ≤ i)

0 (q > i) ∈ b i → gl i ).
(w) = {(1, 1), (3, 3)}, u w = u 1 ⊗ u 3 , S w = 1≤i≤3 K(u 1 ⊗ u i ) = K 1 ⊗ K 3 and ch(S w ) = x 1 (x 1 + x 2 + x 3 ) = S w .
In this case S w is not isomorphic to the Demazure module of the same lowest weight (i) . In general, S w is isomorphic to the Demazure module V (code(w)) of lowest weight code(w) if and only if w is 2143-avoiding. For a general w, there always exists a surjection from S w to V (code(w)).

2

4 Masaki Watanabe
Demazure module V (code(w)) of lowest weight code(w) if and only if w is 2143-avoiding. For a general w, there always exists a surjection from S w to V (code(w)). ✷ For convenience, we slightly generalize the notion of Schubert polynomials and KP modules. For

λ = code(w) ∈ Z n ≥0 (w ∈ S (n)
∞ ) we write S λ = S w and S λ = S w . For a general λ ∈ Z n take k ∈ Z so that λ+k1 ∈ Z n ≥0 and define S λ = x -k1 S λ+k1 and S λ = K -k1 ⊗S λ+k1 (note that this does not depend on the choice of k). We use both permutations and integer vectors as indices of Schubert polynomials and KP modules, but we believe that it will cause no confusion.

A KP filtration of a weight b-module M is a filtration

0 = M 0 ⊂ M 1 ⊂ • • • ⊂ M r = M of b-modules such that each M i /M i-1 is isomorphic to some S λ i (λ i ∈ Z n ).
It is clear from the definition that if M has a KP filtration then ch(M ) is a positive sum of Schubert polynomials.

A criterion for the existence of KP filtration

In [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF], the author investigated characterizations of modules having KP filtrations. The method we used there is similar to the method used in the study of Demazude modules and excellent filtrations ([Pol89], [vdK89]): the use of the theory of highest-weight categories ( [START_REF] Cline | Finite-dimensional algebras and highest weight categories[END_REF]).

In order to develop an analogue of the theory on excellent filtration for KP modules using the highestweight category theory, an important step is to characterize KP modules as projective objects in certain categories of b-modules, as Polo did for Demazure modules in [START_REF] Polo | Variétés de Schubert et excellentes filtrations[END_REF]. We proved in [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF] the following such characterization.

Introduce an order ≤ on Z n as follows. For λ, µ ∈ Z n , take k so that λ + k1, µ + k1 ∈ Z n ≥0 , and define λ ≤ µ ⇐⇒ ℓ(w) = ℓ(v) and w -1 ≥ lex v -1 where w, v ∈ S Example 4.2 Let n = 3. Then the weights λ ∈ Z n which are less than or equal to (0, 2, 1), with respect to the ordering ≤, are (1, 1, 1), (2, 1, 0), (1, 2, 0), (2, 0, 1), (3, 0, 0), (1, 0, 2) and (0, 2, 1). Then it is easy to module (say V (0, 0, 1)), and thus results such as Theorem 4.3 for this kind of KP modules follow from known results on Demazure modules. But in fact there also exist KP modules which are, even as a U (n + )-module, not isomorphic to any Demazure modules: an example is For convenience, we slightly generalize the notion of Schubert polynomials and KP modules. For

S [13254] ∼ = K 2 ⊗ K 4 .
λ = code(w) ∈ Z n ≥0 (w ∈ S (n)
∞ ) we write S λ = S w and S λ = S w . For a general λ ∈ Z n take k ∈ Z so that λ+k1 ∈ Z n ≥0 and define S λ = x -k1 S λ+k1 and S λ = K -k1 ⊗S λ+k1 (note that this does not depend on the choice of k). We use both permutations and integer vectors as indices of Schubert polynomials and KP modules, but we believe that it will cause no confusion.

A KP filtration of a weight b-module M is a filtration

0 = M 0 ⊂ M 1 ⊂ • • • ⊂ M r = M of b-modules such that each M i /M i-1 is isomorphic to some S λ i (λ i ∈ Z n ).
It is clear from the definition that if M has a KP filtration then ch(M ) is a positive sum of Schubert polynomials.

A criterion for the existence of KP filtration

In [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF], the author investigated characterizations of modules having KP filtrations. The method we used there is similar to the method used in the study of Demazude modules and excellent filtrations ([Pol89], [vdK89]): the use of the theory of highest-weight categories ( [START_REF] Cline | Finite-dimensional algebras and highest weight categories[END_REF]).

In order to develop an analogue of the theory on excellent filtration for KP modules using the highestweight category theory, an important step is to characterize KP modules as projective objects in certain categories of b-modules, as Polo did for Demazure modules in [START_REF] Polo | Variétés de Schubert et excellentes filtrations[END_REF]. We proved in [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF] the following such characterization.

Introduce an order ≤ on Z n as follows. For λ, µ ∈ Z n , take k so that λ + k1, µ + k1 ∈ Z n ≥0 , and define λ ≤ µ ⇐⇒ (w) = (v) and w -1 ≥ lex v -1 where w, v ∈ S Example 4.2 Let n = 3. Then the weights λ ∈ Z n which are less than or equal to (0, 2, 1), with respect to the ordering ≤, are (1, 1, 1), (2, 1, 0), (1, 2, 0), (2, 0, 1), (3, 0, 0), (1, 0, 2) and (0, 2, 1). Then it is easy to modules. But in fact there also exist KP modules which are, even as a U (n + )-module, not isomorphic to any Demazure modules: an example is

S [13254] ∼ = K 2 ⊗ K 4 .
Proof: 1. is easy since Ext 1 (M, N ) =

Ext 1 (M i , N ) for any N . 2. also follows from Theorem 4.5, since if 0 → L → M → N → 0 is an exact sequence then we have an exact sequence Ext 1 (M, A) → Ext 1 (L, A) → Ext 2 (N, A) for any A.

2

Using this, we can show that the tensor product of KP modules admits a KP filtration:

, j1 S si 1 ⊗ • • • ⊗ jr S si r ⊗ S v
have a KP filtration for any i 1 , j 1 , . . . , i r , j r and v ∈ S

(n)

∞ . If w ∈ S n , it can be shown, also using the cases of Monk's formula, that for suitable i 1 . . . , i n-1 there exists an exact sequence 0

→ S w → i1 S s1 ⊗ • • • ⊗ in-1 S sn-1 → N → 0 such that N have a KP
filtration whose subquotients are of the form S u (u -1 > lex w -1 ). Then the claim follows by the induction on w using the exact sequence 0 Proof: Let s λ denote the Schur functor corresponding to λ (here we are using the same symbol as Schur functions, but we hope that it would cause no confusion). Then s λ (S w ) is a direct sum component of S ⊗|λ| w , and thus, by Proposition 5.1.1 and Theorem 5.2, have a KP filtration. Thus the claim follows since s λ [S w ] is the character of s λ (S w ). 2

→ S w ⊗ S v → i1 S s1 ⊗ • • • ⊗ in-1 S sn-1 ⊗ S v → N ⊗ S v →
Remark 5.5 The method we used here, i.e. the use of KP modules and highest-weight category theory for the study of Schubert positivity, is of course not only applicable for the proof of the positivity of product and plethystic composition but also for the investigation of other positivity properties of Schubert polynomials. For example, by considering the restriction of KP modules to certain subalgebra of b and using the similar method here (i.e. using highest-weight category theory to reduce the problem to the simplest case S si ), it can be shown that for any w ∈ S

(n)

∞ and I = {i 1 < • • • < i r }, J = {j 1 < • • • < j s }
with I J = {1, . . . , n}, the polynomial S w (z 1 , . . . , z n ) (z k = x p (k = i p ) y q (k = j q ) ) is a positive sum of polynomials of the form S u (x 1 , . . . , x r )S v (y 1 , . . . , y s ). As one of the referees pointed out, this result appears in a work [BS98, Theorem 4.5.4] of Bergeron and Sottile, where the coefficient appearing in the restriction is related to the multiplication coefficients. It would be an interesting problem to investigate module-theoretic counterpart to this relation.

Figure 1 :

 1 Figure 1: Rothe diagram of w = [25143] (corresponding to the positions of the black dots above) is D(w) = {(1, 1), (2, 1), (2, 3), (2, 4), (4, 3)} (shown by the boxes above).

Figure 1 :

 1 Figure 1: Rothe diagram of w = [25143] (corresponding to the positions of the black dots above) is D(w) = {(1, 1), (2, 1), (2, 3), (2, 4), (4, 3)} (shown by the boxes above).

  This corresponds to the fact that Schubert polynomials indexed by permutations are just the Schur polynomials. Example 3.4 If w is 2143-avoiding, then it is known ([Mac91, (1.27)]) that the rows of D(w) are totally (pre)ordered by inclusion. Thus in this case u w is an extremal weight vector in a certain irreducible representation of gl n , and thus S w is isomorphic to a Demazure module: i.e. the U(b)-module generated by an extremal weight vector in a irreducible representation of gl n . Note that this corresponds to the result of Lascoux and Schutzenberger ([LS89, Theorem 5], [Las03, Corollary 10.5.2]) that Schubert polynomials with 2143-avoiding indices are equal to certain key polynomials. On the other hand, consider w = [2143]. Then D

Figure 2 :

 2 Figure 2: Rothe diagram of a Grassmannian permutation w = [136245]. Here uw = (u2 ∧ u3) ⊗ u3 ⊗ u3 is a lowest vector in an irreducible representation of gl 3 with lowest weight (0, 1, 3).

∞

  are defined by code(w) = λ + k1 and code(v) = µ + k1. Here ≥ lex denotes the lexicographic ordering on permutations. For λ ∈ Z n ≥0 , let C ≤λ be the category of all weight b-modules whose weights are all ≤ λ.

  The modules S λ and S * ρ-λ ⊗ K ρ are in C ≤λ and give the projective cover and the injective hull of K λ in C ≤λ respectively.

Figure 2 :

 2 Figure 2: Rothe diagram of a Grassmannian permutation w = [136245]. Here uw = (u2 ∧ u3) ⊗ u3 ⊗ u3 is a lowest vector in an irreducible representation of gl 3 with weight (0, 1, 3).

∞

  are defined by code(w) = λ + k1 and code(v) = µ + k1. Here ≥ lex denotes the lexicographic ordering on permutations. For λ ∈ Z n ≥0 , let C ≤λ be the category of all weight b-modules whose weights are all ≤ λ.

  Theorem 4.1 ([Wat14a, Proposition 6.4]) The modules S λ and S * ρ-λ ⊗ K ρ are in C ≤λ and give the projective cover and the injective hull of K λ in C ≤λ respectively.

  Theorem 5.2 ([Wat14b, Theorem 4.1]) For any w, v ∈ S (n) ∞ , the module S w ⊗ S v have a KP filtration. Outline of Proof: First check the case w = s i (this corresponds to Monk's formula for Schubert polynomials) by explicitly constructing a filtration. Then we see that S si ⊗ S sj ⊗ • • • ⊗ S s k ⊗ S v have a KP filtration for any i, j, . . . , k and v ∈ S (n) ∞ , and by Proposition 5.1.1

∞

  0 and Proposition 5.1.2. For a general w the claim follows by letting n large, since for n > n, b n -modules can be naturally identified with b n -modules on which e ij (j > n) acts by zero and under this identification the b n -module S w (w ∈ S ) corresponds to the b n -module S w . 2 Remark 5.3 As we pointed in [Wat14a] and [Wat14b], Theorem 5.2 gives a new description of generalized Littlewood-Richardson coefficients, in a manner somewhat similar to triple-intersection formula in the geometric theory of Schubert calculus: for u, v, w ∈ S n , if the polynomial S u S v is expanded into a sum of Schubert polynomials, the coefficient of S w is equal to the dimension of Hom b (S u ⊗ S v , S * w0w ⊗ K ρ ) ∼ = Hom b (S u ⊗ S v ⊗ S w0w , K ρ ). As a corollary of Theorem 5.2, we can show that the plethystic composition of a Schur function with a Schubert polynomial is Schubert-positive: Corollary 5.4 ([Wat14b, Corollary 4.3]) Let λ be a partition and s λ be the corresponding Schur function. Let w ∈ S (n) ∞ and write S w as a sum of monomials as S w = x α + x β + • • • . Then s λ [S w ] := s λ (x α , x β , . . .) is a positive sum of Schubert polynomials.

(i) The KP module S[2143] in this example is, if not seen as a U (b)-module but as a U (n + )-module, isomorphic to a Demazure module (say V (0, 0, 1)), and thus results such as Theorem 4.3 for this kind of KP modules follow from known results on Demazure

If 0 → L → M → N → 0 is exact and M and N have KP filtrations, then L also has a KP filtration.

check that the projective cover of K (0,2,1) in C ≤(0,2,1) is isomorphic to the KP module S (0,2,1) = S [1432] , which has character S [1432] = x 2 1 x 2 + x 2 1 x 3 + x 1 x 2 2 + x 1 x 2 x 3 + x 2 2 x 3 . In the proof of this characterization, we used the following structural result on KP modules. The current proof of the following result in [START_REF] Watanabe | Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz's modules realizing Schubert polynomials as their characters[END_REF] uses a long case-by-case check, and we hope that there is a more elegant proof.

For

The following theorem states that the annihilator of u w is in fact generated by these elements: 1. M has a KP filtration.

Ext

Applications

In this section we present two applications of Theorem 4.5. One application is the proof of the fact that the tensor product S w ⊗ S v of KP modules admits a KP filtration. This can be seen as a module-theoretic counterpart of the classical fact that the product S w S v of Schubert polynomials is a positive sum of Schubert polynomials, and it also gives a new proof to this fact. Another application is the Schubertpositivity of the "plethystic composition" s λ [S w ] of a Schur function with a Schubert polynomial. First we prepare some useful corollaries from Theorem 4.5: