Kra\'skiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials

In their 1987 paper Kra\'skiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of KP modules always has a KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely $\mathcal{S}_w \otimes S^d(K^i)$ and $\mathcal{S}_w \otimes \bigwedge^d(K^i)$, corresponding to Pieri and dual Pieri rules for Schubert polynomials.


Introduction
Schubert polynomials are one of the main subjects in algebraic combinatorics. One of the tools for studying Schubert polynomials is the modules introduced by Kraśkiewicz and Pragacz. These modules, which here we call KP modules, are modules over the upper triangular Lie algebra and have the property that their characters with respect to the diagonal matrices are Schubert polynomials.
It is known that a product of Schubert polynomials is always a positive sum of Schubert polynomials. The previously known proof for this positivity property uses the geometry of the flag variety. In [8] the author showed that the tensor product of two KP modules always has a filtration by KP modules and thus gave a representation-theoretic proof for this positivity. Although the proof there does not give explicit constructions for the KP filtrations, it may provide a new viewpoint for the notorious Schubert-LR problem asking for a combinatorial positive rule for the coefficient in the expansion of products of Schubert polynomials into a sum of Schubert polynomials.
There are some cases where the expansions of products of Schubert polynomials are explicitly known. Examples of such cases include the Pieri and the dual Pieri rules for Schubert polynomials ( [1], [6], [7], [10]). These are the cases where one of the Schubert polynomials is a complete symmetric function h d (x 1 , . . . , x i ) or an elementary symmetric function e d (x 1 , . . . , x i ). The purpose of this paper is to investigate the structure of tensor product modules corresponding to these products and to give explicit constructions of KP filtrations for these modules.
The structure of this paper is as follows. In Section 2 we prepare some definitions and results on Schubert polynomials and KP modules. In Section 3 we review the Pieri and the dual Pieri rules for Schubert polynomials. In Section 4 we give explicit constructions for KP filrartions of the corresponding tensor product modules S w ⊗ S d (K i ) and S w ⊗ d (K i ). In Section 5 we give a proof of the main result.

Preliminaries
Let N be the set of all positive integers. By a permutation we mean a bijection from N to itself which fixes all but finitely many points. The graph of a permutation w is the set {(i, w(i)) : i ∈ N} ⊂ N 2 . 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)}. For a permutation w and positive integers p < q, if ℓ(wt pq ) = ℓ(w) + 1 we write wt pq ⋗ w. It is well known that this condition is equivalent to saying that w(p) < w(q) and there exists no p < r < q satisfying w(p) < w(r) < w(q). For a permutation w let For a polynomial f = f (x 1 , x 2 , . . .) and i ∈ N define ∂ i f = f −sif xi−xi+1 . For a permutation w we can assign its Schubert polynomial S w ∈ Z[x 1 , x 2 , . . .] which is recursively defined by · · · x n−1 if w(1) = n, w(2) = n − 1, . . . , w(n) = 1 and w(i) = i (i > n) for some n, and Hereafter let us fix a positive integer n. Let Note that if w ∈ S (n) then I(w) ⊂ {1, . . . , n} × N. Let K be a field of characteristic zero. Let b = b n denote the Lie algebra of all n × n upper triangular matrices over K. For a b-module M and λ = (λ 1 , . . . , λ n ) ∈ Z n , let M λ = {m ∈ M : hm = λ, h m (∀h = diag(h 1 , . . . , h n ))} where λ, h = i λ i h i . If M is a direct sum of these M λ and these M λ are finite dimensional then we say that M is a weight module and we define ch(M ) = λ dim M λ x λ where x λ = x λ1 1 · · · x λn n . For 1 ≤ i ≤ j ≤ n let e ij ∈ b be the matrix with 1 at the (i, j)-th position and all other coordinates 0.
Let U be a vector space spanned by a basis {u ij : 1 ≤ i ≤ n, j ∈ N}. Let T = ∞ d=0 d U . The Lie algebra b acts on U by e pq u ij = δ iq e pj and thus on T .
For w ∈ S (n) let u w = (i,j)∈I(w) u ij ∈ ℓ(w) U ⊂ T . The Kraśkiewicz-Pragacz module S w (or the KP module for short) associated to w is the b-submodule of ℓ(w) U ⊂ T generated by u w . In [5] Kraśkiewicz and Pragacz showed the following: Theorem 2.1 ([5, Remark 1.6 and Theorem 4.1]). S w is a weight module and ch(S w ) = S w .
3 Pieri and dual Pieri rules for Schubert polynomials Note that the condition for X i,d (w) (resp. Y i,d (w)) implies that q 1 , . . . , q d (resp. p 1 , . . . , p d ) are all different.
Theorem 3.2 (Conjectured in [1] and proved in [10], also appears with different formulations in [6] and [7]). We have where h d and e d denote the complete and elementary symmetric functions respectively.
Note here that the permutation ζ ∈ X i,d (w) (or Y i,d (w)) in fact uniquely determines its decomposition into transpositions satisfying the conditions in Definition 3.1. So we can write, without ambiguity, for example "for ζ = t p1q1 · · · t p d q d ∈ X i,d (w) define (something) as (some formula involving p j and q j )". Hereafter if we write such we will always assume the conditions in Definition 3.1.

Explicit Pieri and dual Pieri rules for KP modules
The author showed in [8] that the tensor product of KP modules always has a KP filtration. Since S d (K i ) and In this section we give explicit constructions for these filtrations. For positive integers p ≤ q we define an operator e ′ qp acting on T as e ′ qp (u a1b1 ∧ u a2b2 ∧ · · · ) = k (· · · ∧ δ pb k u a k q ∧ · · · ). Let these operators act on T ⊗ S d (K i ) and T ⊗ d (K i ) by applying them on the left-hand side tensor component.
For a permutation z and positive integers p < q let m pq (z) = #{r > q : z(p) < z(r) < z(q)} and m ′ qp (z) = #{r < p : z(p) < z(r) < z(q)}. For where w j = wt p1q1 · · · t pj−1qj−1 as in Definition 3.1. Note that these are also welldefined even if some q j are greater than n, since in such a case m pj qj (w j ) = 0. Note also that the products of the operators e pj qj above are well-defined since the operators e pj qj (p j ≤ i, q j > i) commute with each others. Also, for such ζ, Note that the order in the product symbol does not matter since the operators e ′ qj pj commute. Let < lex and < rlex denote the lexicographic and reverse lexicographic orderings on permutations respectively, i.e. for permutations u and v, u < if there exists a k such that u(j) = v(j) for all j < k (resp. j > k) and u(k) < v(k).
The proof for this proposition is given in the next section. Here we first see that Proposition 4.1 gives desired filtrations.
For a b-module M and elements x, y, . . . , z ∈ M let x, y, . . . , z denote the submodule generated by these elements. Consider the sequence of submodules ) are all the elements ordered increasingly by the lexicographic (resp. reverse lexicographic) ordering of (wζ) −1 . From the proposition we see that there are surjections v ζ1 , · · · , v ζj / v ζ1 , · · · , v ζj−1 ։ S wζj induced from φ ζj . Thus we have respectively, where the last equalities are by Proposition 3.2. So the equalities must hold everywhere. Thus v ζ : ζ ∈ X i,d (w) (resp. Y i,d (w)) = S w ⊗ S d (K i ) (resp. S w ⊗ d (K i )) and the surjections above are in fact isomorphisms. So, in conclusion, we get from Proposition 4.1 the following: Define v ζ and φ ζ as above. Then M is generated by {v ζ : ζ ∈ X i,d (w) (resp. Y i,d (w))} as a b-module and gives a KP filtration of M , where ζ 1 , ζ 2 , . . . ∈ X i,d (w) (resp. Y i,d (w)) are all the elements ordered increasingly by the lexicographic (resp. reverse lexicographic) ordering of (wζ) −1 . The explicit isomorphism v ζ1 , · · · , v ζj / v ζ1 , · · · , v ζj−1 ∼ = S wζj is given by φ ζj defined above. Remark 4.3. In [9] we related KP modules with the notion of highest weight categories ( [3]) as follows. For Λ ′ ⊂ Z n let C Λ ′ be the category of weight b nmodules whose weights are all in Λ ′ . Then if Λ ′ is an order ideal with respect to a certain ordering on Z n then C Λ ′ has a structure of highest weight category whose standard objects are KP modules. One of the axioms for highest weight categories requires that the projective objects should have filtrations by standard objects.
It can be shown that the projective cover of the one dimensional b n -module K λ with weight λ = (λ 1 , . . . , λ n ) ∈ Z n ≥0 in the category C Z n ≥0 is given by S λ1 (K 1 ) ⊗ · · · ⊗ S λn (K n ). Thus Theorem 4.2 gives a proof to the fact that the indecomposable projective modules in C Z n ≥0 have KP filtrations, which leads to a different proof from the one in [9, §3] for the axiom mentioned above (we do not need these results about highest weight structure for b-modules in the proof of Theorem 4.2 which will be done below).

Proof of Proposition 4.1
Lemma 5.1. Let w ∈ S (n) and i ≥ 1. For p, p ′ ≤ i and q, q ′ > i such that Proof. This is essentially the same as [9,Lemma 5.8].
where w a = wt p1q1 · · · t pa−1qa−1 as in Definition 3.1. Proof. We show the case ζ ∈ X i,d (w): the other case can be treated similarly. First note that p 1 , . . . , p a−1 = p a by the hypothesis. Also, as we have remarked before, q 1 , . . . , q a are all different. Thus the proof is now reduced to the following lemma.
Lemma 5.3. Let p < q, p ′ < q ′ and suppose Proof. Let us begin with a simple observation: suppose there exist two rectangles R 1 and R 2 with edges parallel to coordinate axes. Suppose that no two edges of these rectangles lie on the same line. Then, checking all the possibilities we see that #(NW and SE corners of R 1 lying inside R 2 ) − #(NE and SW corners of R 1 lying inside R 2 ) = #(NW and SE corners of R 2 lying inside R 1 ) − #(NE and SW corners of R 2 lying inside R 1 ).
First consider the case R 1 = [p, q]×[w(p), w(q)] and R 2 = [p ′ , q ′ ]×[w(p ′ ), w(q ′ )] in the observation above. wt p ′ q ′ t pq ⋗ wt p ′ q ′ ⋗ w implies that the first term in the left-hand side and the second term in the right-hand side vanish (here the coordinate system is taken so that points with smaller coordinates go NW). Thus all the terms must vanish. In particular the first term on the right-hand side vanishes and thus wt pq ⋗ w.
We have shown that none of the points (p, w(p)), (p, w(q)), (q, w(p)) and (q, w(q)) lie in [p ′ , q ′ ] × [w(p ′ ), w(q ′ )]. Since m pq (w) (resp. m p ′ q ′ (w)) is the number of points of the graph of w lying inside the rectangle for M ≫ 0 and the graphs of w and wt p ′ q ′ differ only at the vertices of the rectangle applying the observation to these rectangles shows the remaining claims.
Proof of Proposition 4.1.
Next consider the case w(p 1 ) = w(p ′ 1 ) and a > 1. In this case we see (w) u w = 0 since w(p a ) = w a (p a ) > w(p 1 ) = w(p ′ 1 ). Next consider the case w(p 1 ) = w(p ′ 1 ), a = 1 and q 1 < q ′ 1 . Then since wt p1q1 , wt p ′ 1 q ′ 1 ⋗ w it follows that w(q ′ 1 ) < w(q 1 ). So again by Lemma 5.1 we see So the only remaining summands in ( * ) are the ones with (p 1 , q 1 ) = (p ′ 1 , q ′ 1 ) and a = 1, i.e. p σ(1) = p 1 . It is easy to see that the sum of such summands is a nonzero multiple of the sum of terms with σ(1) = 1. If σ(1) = 1 we have, by the latter part of Lemma 5.1, So, working inductively on d (using wt p1q1 , t p2q2 · · · t p d q d and t p ′ or if ζ ′ = ζ and σ = id, and • if ζ ′ = ζ and σ = id then it is a nonzero multiple of u wζ . This finishes the proof for X i,d (w). Proof for Y i,d (w): This proceeds much similarly to the previous case. Here instead of ( * ) we use where E j = e mp j q j (wj ) pj qj We assume (wζ) −1 ≤ rlex (wζ ′ ) −1 . Fix σ and take 1 ≤ a ≤ d minimal with q ′ a = q ′ σ −1 (1) . By an argument similar to the above, it suffices to show that u p1q ′ a ∧ e (w) u w is zero unless a = 1 and (p ′ 1 , q ′ 1 ) = (p 1 , q 1 ), and in a such case it is a nonzero multiple of u wtp1q1 .