Symmetry, Integrability and Geometry: Methods and Applications Skew Divided Difference Operators and Schubert Polynomials ⋆

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.


Introduction
In this paper we study the skew divided difference operators with applications to the "Littlewood-Richardson problem" in the Schubert calculus. By the Littlewood-Richardson problem in the Schubert calculus we mean the problem of finding a combinatorial rule for computing what one calls the structural constants for Schubert polynomials. These are the structural constants c w uv , u, v, w ∈ S n , of the ring P n /I n , where P n is the polynomial ring Z[x 1 , . . . , x n ] and I n is the ideal of P n generated by the symmetric polynomials without constant terms, with respect to its Z-free basis consisting of the classes of Schubert polynomials S w , w ∈ S n . Namely, the constants c w uv are defined via the decomposition of the product of two Schubert polynomials S u and S v modulo the ideal I n : The symbol w v for w, v ∈ S n , here and after, means that w dominates v with respect to the Bruhat order on the symmetric group S n (see, e.g., [11, p. 6]). Formula (1.2) is reduced to the classical Leibnitz rule in the case when w = (i, i + 1) is a simple transposition: One of the main applications of the skew divided difference operators is an elementary and transparent algebraic proof of the Monk formula for Schubert polynomials (see [11, equation (4.15)]).
Our interest to the skew divided difference operators is based on their connection with the structural constants for Schubert polynomials. More precisely, if w, v ∈ S n , and w v, then 3) The polynomial ∂ w/v (S u ) is a homogeneous polynomial in x 1 , . . . , x n of degree l(u) + l(v) − l(w) with integer coefficients. We make a conjecture that in fact (1.4) i.e. the polynomial ∂ w/v (S u ) has nonnegative integer coefficients. In the case l(u) + l(v) = l(w), this conjecture follows from the geometric interpretation of the structural constants c w uv as the intersection numbers for Schubert cycles. For general u, v, w ∈ S n the conjecture is still open.
In Section 8 we prove conjecture (1.4) in the simplest nontrivial case (see Theorem 1) when w and v are connected by an edge in the Bruhat order on the symmetric group S n . In other words, if w = vt ij , where t ij is the transposition that interchanges i and j, and l(w) = l(vt ij )+ 1. It is well-known [11, p. 30] that in this case the skew divided difference operator ∂ w/v coincides with operator ∂ ij , i.e. ∂ w/v = ∂ ij . Our proof employs the generating function for Schubert polynomials ("Schubert expression" [5,3,4]) in the nilCoxeter algebra.
In Section 9 we consider another application of the skew divided difference operators, namely, we give an explicit (but still not combinatorial) formula for structural constants for Schubert polynomials in terms of weighted paths in the Bruhat order with weights taken from the nil-Coxeter algebra (see Theorem 2).
It is well known that there are several equivalent ways to define the skew Schur functions, see e.g., [11,12]. Apart from the present paper, a few different definitions of skew Schubert polynomials have been proposed in [1,10] and [2]. These definitions produce, in general, different polynomials.

Skew Schur functions
In this Section we review the definition and basic properties of the skew Schur functions. For more details and proofs, see [12,Chapter I,Section 5]. The main goal of this Section is to arise a problem of constructing skew Schubert polynomials with properties "similar" to the those for skew Schur functions (see properties (2.2)-(2.5) below).
Let X n = (x 1 , . . . , x n ) be a set of independent variables, and λ, µ be partitions, µ ⊂ λ, l(λ) ≤ n. Definition 1. The skew Schur function s λ/µ (X n ) corresponding to the skew shape λ − µ is defined to be where h k := h k (X n ) is the complete homogeneous symmetric function of degree k in the variables X n = (x 1 , . . . , x n ).
Below we list the basic properties of skew Schur functions: a) Combinatorial formula: where summation is taken over all semistandard tableaux T of the shape λ − µ with entries not exceeding n; here w(T ) is the weight of the tableau T (see, e.g., [12, p. 5]), and x w(T ) := x w 1 1 x w 2 2 · · · x wn n . b) Connection with structural constants for Schur functions: where the coefficients c λ µν (the structural constants, or the Littlewood-Richardson numbers) are defined through the decomposition c) Littlewood-Richardson rule: where Tab 0 (λ − µ, ν) is the set of all semistandard tableaux T of shape λ − µ and weight ν such that the reading word w(T ) of the tableaux T (see, e.g., [12, Chapter I, Section 9]) is a lattice word (ibid). Thus,

Divided difference operators
Definition 2. Let f be a function of x and y (and possibly other variables), the divided difference operators ∂ xy is defined to be The operator ∂ xy takes polynomials to polynomials and has degree −1. On a product f g, ∂ xy acts according to the Leibniz rule where s xy interchanges x and y.
It is easy to check the following properties of divided difference operators ∂ xy : The next step is to define a family of divided difference operators ∂ i , 1 ≤ i ≤ n − 1, which act on the ring of polynomials in n variables. Let x 1 , x 2 , . . . , x n be independent variables, and let For each i, 1 ≤ i ≤ n − 1, let be the divided difference operator corresponding to the simple transposition s i = (i, i + 1) which interchanges x i and x i+1 .
Each ∂ i is a linear operator on P n of degree −1. The divided difference operators ∂ i , 1 ≤ i ≤ n − 1, satisfy the following relations Let w ∈ S n be a permutation; then w can be written as a product of simple transpositions For any sequence a = (a 1 , . . . , a p ) of positive integers, let us define ∂ a = ∂ a 1 · · · ∂ ap .

Proposition 1 ([11], Chapter II).
i) If a sequence a = (a 1 , . . . , a p ) is not reduced, i.e. not a reduced decomposition of any From Proposition 1, ii) follows that one can define ∂ w = ∂ a unambiguously, where a is any reduced decomposition of w.

Schubert polynomials
In this section we recall the definition and basic properties of the Schubert polynomials introduced by A. Lascoux and M.-P. Schützenberger. Further details and proofs can be found in [11].

Definition 3 ([9]
). For each permutation w ∈ S n the Schubert polynomial S w is defined to be where w 0 is the longest element of S n .
ii) Let v, w ∈ S n . Then iii) The Schubert polynomials S w , w ∈ S n , form a Z-linear basis in the space F n , where iv) The Schubert polynomials S w , w ∈ S n , form an orthogonal basis with respect to the pairing , 0 : Let m > n and let i : S n ֒→ S m be the natural embedding. Then

Skew divided difference operators
The skew divided difference operators ∂ w/v , w, v ∈ S n , were introduced by I. Macdonald [11, Chapter II]. Let w, v ∈ S n , and w v with respect to the Bruhat order on the symmetric group S n . In other words, if a = (a 1 , . . . , a p ) is a reduced decomposition of w then there exists a subsequence b ⊂ a such that b is a reduced decomposition of v (for more details, see, e.g., [11, equation (1.17)].

Definition 4 ([11]
). Let v, w ∈ S n , and w v with respect to the Bruhat order, and a = (a 1 , . . . a p ) ∈ R(w). The skew divided difference operator ∂ w/v is defined to be One can show (see, e.g., [11, p. 29]) that Definition 4 is independent of the reduced decomposition a ∈ R(w).
Below we list the basic properties of the skew divided difference operators ∂ w/v . For more details and proofs, see, e.g., [11]. The statement iv) of Proposition 3 below seems to be new.

Proposition 3.
i) Let f, g ∈ P n , w ∈ S n , then More generally, ii) Let f, g ∈ P n , u, w ∈ S n , and w u with respect to the the Bruhat order. Then , and t = t ij is the transposition that interchanges i and j and fixes all other elements of [1, n]. Then Then where c w uv are the structural constants for the Schubert polynomials S w , w ∈ S n ; in other words, where I n is the ideal generated by the elementary symmetric functions e 1 (x 1 , . . . , x n ), . . . , e n (x 1 , . . . , x n ).
Proof . We refer the reader to [11, p. 30] for proofs of statements i)-iii). iv) To prove the identity (5.5), we will use the formula (5.2) and the following result due to I. Macdonald [11, equation (5.7)]): where for each permutation w ∈ S n , ǫ(w) = (−1) l(w) is the sign (signature) of w.
Using the generalized Leibnitz formula (5.2), we can write the LHS (5.7) as follows: Comparing the RHS of (5.7) and that of (5.8), we see that To finish the proof of equality (5.5), it remains to apply the following formula [11, equation (2.12)]: v) We consider formula (5.6) as a starting point for applications of the skew divided difference operators to the problem of finding a combinatorial formula for the structural constants c w uv ("Littlewood-Richardson problem" for Schubert polynomials, see Section 2). Having in mind some applications of (5.6) (see Sections 8 and 9), we reproduce below the proof of (5.6) given by I. Macdonald [11, p. 112]. It follows from Proposition 2 ii) and Proposition 3 i) that In the latter sum the only nonzero term appears when v 1 = v. Hence, It is well-known (and follows, for example, from Proposition 2, i) and ii)) that for each v, w ∈ S n More generally, we make the following conjecture.
We used the following formulae for Schubert polynomials: , and . Let us note that 6 Analog of skew divided differences in the Bracket algebra In this Section for each v, w ∈ S n we construct the element [w/v] in the Bracket algebra E 0 n which is an analog of the skew divided difference operators ∂ w/v . The Bracket algebra E 0 n was introduced in [4]. By definition, the Bracket algebra E 0 n (of type A n−1 ) is the quadratic algebra (say, over Z) with generators [ij], 1 ≤ i < j ≤ n, which satisfy the following relations (i) [ij] 2 = 0, for i < j; ( For further details, see [4,6]. Note that [ij] → ∂ ij , 1 ≤ i < j ≤ n, defines a representation of the algebra E 0 n in P n . Now, let v, w ∈ S n , and w v with respect to the Bruhat order on S n . Let a ∈ R(w) be a reduced decomposition of w. We define the element [w/v] in the Bracket algebra E 0 n to be Note that the right-hand side of the definition of [w/v] can be interpreted inside the crossed product of E 0 n by S n (which is also called a skew group algebra in this case) with respect to the action of S n on E 0 n defined by eventually giving an element of E 0 n . Remark 1. Let w, v ∈ S n , and w v. One can show that the element [w/v] ∈ E 0 n is independent of the reduced decomposition a ∈ R(w).

Conjecture 2. The element [w/v] ∈ E 0
n can be written as a linear combination of monomials in the generators [ij], i < j, with nonnegative integer coefficients.

Skew Schubert polynomials
Definition 5. Let v, w ∈ S n , and w v with respect to the Bruhat order. The skew Schubert polynomial S w/v is defined to be Example 3. a) Let w = s 1 s 2 s 3 s 1 ∈ S 4 , and v = s 1 ∈ S 4 . Then v −1 w 0 = s 2 s 1 s 3 s 2 s 1 , w −1 w 0 = s 2 s 1 , and It is clear that if w, v ∈ S n , and w v, then S w/v is a homogeneous polynomial of degree n 2 − l(w) + l(v) with integer coefficients. It would be a corollary of Conjecture 1 that skew the Schubert polynomials have in fact positive integer coefficients.

Proposition 4.
i) Let v ∈ S n , and w 0 ∈ S n be the longest element. Then Proof of (7.2) follows from (5.5) and (7.1).
It is an interesting task to find the Monk formula for skew Schubert polynomials, in other words, to describe the decomposition of the product (x 1 +· · ·+x r )S w/v , w, v ∈ S n , 1 ≤ r ≤ n−1, in terms of Schubert polynomials. 1 · · · x mn n ⊗ e w , m 1 , . . . , m n ∈ N, w ∈ S n , in Z[x 1 , . . . , x n ] ⊗ Z N C n . Similar remarks apply to similar notation below.
The study of action of divided difference operators ∂ ij , 1 ≤ i < j ≤ n, on the Schubert expression S (n+1) is the main step of our proof of Conjecture 1 for the skew divided difference operators corresponding to the edges in the Bruhat order on the symmetric group S n+1 . In exposition we follow to [5,3,4]. Definition 6. The nilCoxeter algebra N C n is the algebra (say, over Z) with generators e i , 1 ≤ i ≤ n, which satisfy the following relations (i) e 2 i = 0, for 1 ≤ i ≤ n, (ii) e i e j = e j e i , for 1 ≤ i, j ≤ n, |i − j| > 1, (iii) e i e j e i = e j e i e j , for 1 ≤ i, j ≤ n, |i − j| = 1.
For each w ∈ S n+1 let us define e w ∈ N C n to be e w = e a 1 · · · e ap , where (a 1 , . . . , a p ) is any reduced decomposition of w. The elements e w , w ∈ S n+1 , are well-defined and form a Z-basis in the nilCoxeter algebra N C n . Now we are going to define the Schubert expression S (n+1) which is a noncommutative generating function for the Schubert polynomials. Namely, The basic property of the Schubert expression S (n+1) is that it admits the following factorization [5]: Now we are ready to formulate and prove the main result of this Section, namely, the following positivity theorem: Proof . Our starting point is the Lemma below which is a generalization of the Statement 4.19 from Macdonald's book [11]. Before to state the Lemma, we need to introduce a few notation.
Lemma 1. The coefficients d w u 1 ,...,up defined above, are non-negative integers. The proof of Lemma proceed by induction on l(u p ), and follows very close to that given in [11]. We omit details.
It follows from the Lemma above that it is enough to prove Theorem 1 only for the transposition (i, j) = (1, n). Thus, we are going to prove that ∂ 1n S w ∈ N[x 1 , . . . , x n ]. For this goal, let us consider the Schubert expression S (n+1) = A 1 (x 1 )A 2 (x 2 ) · · · A n (x n ), see (8.1). We are going to prove that where α w (x) ∈ N[x 1 , . . . , x n ] for all w ∈ S n+1 . Using the Leibniz rule (3.2), we can write First of all, The next observation is c k x k , where c k ∈ N C n , c 0 = 1, then , as it was claimed. Hence, Thus, it is enough to prove that the difference belongs to the set N[x 1 , . . . , x n ][N C n ]. We will use the following result (see [5,3]): Thus, using a simple observation that Let us denote the sum over i in (5.3) by G(x 1 , . . . , x n ). It is clear that Thus the difference   where S (n) denotes the Schubert expression. Indeed, The next step is to compute η ∂ w/v S using the following lemma, which is obtained by repetitive use of the generalized Leibniz rule (5.3).