Back stable Schubert calculus

We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman-Greene insertion algorithm; 5) the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case; 6) equivariant Pieri rules for the homology of the infinite Grassmannian; 7) homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

See Section 10. In this paper we study the Schubert calculus of Gr and Fl. The infinite flag variety Fl is the union of finite-dimensional flag varieties, and any product ξ x ξ y of two Schubert classes ξ x , ξ y ∈ H * (Fl) can be computed within some finite-dimensional flag variety. Naively, as some subset of the authors had mistakenly assumed, no interesting and new phenomena would arise in the infinite case. To the contrary, in this article we find entirely new behavior that have no classical counterpart.
1.2. Back stable Schubert polynomials. Polynomial representatives for the Schubert classes in the cohomology H * (Fl n ) of a finite flag variety Fl n , called Schubert polynomials S w , were defined by Lascoux and Schützenberger [LaSc82], following work of Bernstein, Gelfand, and Gelfand [BGG] and Demazure [Dem].
The focus of this work is on the limits of Schubert polynomials called back stable Schubert polynomials ← − S w := lim p→−∞ q→∞ S w (x p , x p+1 , . . . , x q ), for w ∈ S Z , the group of permutations of Z moving finitely many elements. Two of us (T. L. and M. S.) first learned of this construction from Allen Knutson [Knu]. Anders Buch [Buc] was also aware of how to back stabilize (double) Schubert polynomials. Finally, one of us (S.-J. Lee) found them on his own independently. These polynomials form a basis of the ring ← − R := Λ ⊗ Q[. . . , x −1 , x 0 , x 1 , . . .] where Λ denotes the symmetric functions in . . . , x −1 , x 0 . Under an isomorphism between ← − R and the cohomology of Fl, we show in Theorem 10.2 that back stable Schubert polynomials represent Schubert classes of Fl.
The first of the new phenomena we find is the "coproduct formula" (Theorem 3.16) Stanley symmetric function, S v is an ordinary Schubert polynomial, and w . = uv is a length-additive factorization with v a permutation not using the reflection s 0 . Traditionally, the Stanley symmetric functions F w [Sta] are obtained from ordinary Schubert polynomials S w by taking a forward limit. We show that Stanley functions can be obtained from back stable Schubert polynomials by an algebra homomorphism, contrary to the map from finite Schuberts to forward-limit Stanleys, which is not multiplicative. This is closely related to, and explains, a formula of Li [Li].

Double Stanley symmetric functions. Back stable double Schubert polynomials
← − S w (x; a) can also be defined (though the existence of the limit is less clear, see Proposition 4.3). They also satisfy (Theorem 4.14) the same kind of coproduct formula as the non-doubled version, with the double Stanley symmetric functions F w (x||a) replacing F w and double Schubert polynomials replacing the usual finite Schubert polynomials. The elements F w (x||a) lie in the ring Λ(x||a) of double symmetric functions, which is the polynomial Q[a] = Q[. . . , a −1 , a 0 , a 1 , . . . ]-algebra generated by the double power sums p k (x||a) := i≤0 x k i − i≤0 a k i . As far as we are aware, the symmetric functions F w (x||a) are novel. When w is 321-avoiding, the double Stanley symmetric function is equal to the skew double Schur function which was studied by Molev [Mol09]; see Proposition A.2.
One of our main theorems (Theorem 4.20) is a proof that the double Edelman-Greene coefficients j w λ (a) ∈ Q[a] given by the expansion of double Stanley symmetric functions F w (x||a) = λ j w λ (a)s λ (x||a) into double Schur functions s λ (x||a), are positive polynomials in certain linear forms a i − a j . The Edelman-Greene coefficients j w λ (0) are known to be positive by the influential works of Edelman and Greene [EG] and Lascoux and Schützenberger [LaSc85]. We also obtain an explicit combinatorial formula for j w λ (a) in the special case that corresponds to the "equivariant homology Pieri rule" of the infinite flag variety. Molev [Mol09] has given a combinatorial rule for the expansion coefficients of skew double Schurs into double Schurs (that is, for j w λ (a) where w is 321-avoiding) but it does not exhibit the above positivity.
1.4. Bumpless pipedreams. We introduce a combinatorial object called bumpless pipedreams, to study the monomial expansion of back stable double Schubert polynomials. These are pipedreams where pipes are not allowed to bump against each other, or equivalently, the "bumping" or "double elbow tile" is forbidden: Using bumpless pipedreams, we obtain: • An expansion for double Schubert polynomials S w (x; a) in terms of products of binomials (x i − a j ). Our formula is different from the classical pipe-dream formula of Fomin and Kirillov [FK] for double Schubert polynomials: our formula is obviously back stable. Hence we also obtain such an expansion for back stable double Schubert polynomials.
• A positive expression for the coefficient of s λ (x||a) in ← − S(x; a) (Theorem 5.11) • A new combinatorial interpretation of Edelman-Greene coefficients j w λ (0) as the number of certain EG-pipedreams (Theorem 5.14). Our bumpless pipedreams are a streamlined version of the interval positroid pipedreams defined by Knutson [Knu14]. Heuristically, our formula for ← − S w (x; a) is obtained by "pulling back" a Schubert variety in Fl to various Grassmannians where it can be identified (after equivariant shifts) with graph Schubert varieties, a special class of positroid varieties. This connects our work with that of Knutson, Lam, and Speyer [KLS], who identified the equivariant cohomology classes of positroid varieties with affine double Stanley symmetric functions.
When presenting our findings we were informed by Anna Weigandt [Wei] that Lascoux's use [Las02] of alternating sign matrices (ASMs) in a formula for Grothendieck polynomials, is very close to our pipedreams; ours correspond to the subset of reduced ASMs. Our construction has the advantage that the underlying permutation is evident; in the ASM one must go through an algorithm to extract this information. Lascoux's ASMs naturally compute in K-theory rather than in cohomology.
1.5. NilHecke algebra and Peterson subalgebra. Our constructions are fully compatible with torus-equivariance: the localization of a Schubert class ξ w at a T Z -fixed point v ∈ S Z ⊂ Fl is equal to a specialization ← − S w (va; a) of the back stable double Schubert polynomial. Using torus-equivariant localization, we construct actions of the infinite nilHecke ring A ′ on H * T Z (Fl); this is an infinite rank variation of the results of Kostant and Kumar [KK].
As somewhat of a surprise, we are able to construct (Theorem 8.8) a subalgebra P ′ ⊂ A ′ that is an analogue of the Peterson subalgebra in the affine case. Peterson [Pet] constructed this subalgebra of an affine nilHecke algebra as an algebraic model for the equivariant homology H * T ( Gr n ) of the affine Grassmannian Gr n . Our construction in the infinite case is unexpected because the infinite symmetric group S Z is not an affine Coxeter group. Nevertheless, we are able to construct elements in A ′ that behave like translation elements in affine Coxeter groups.
1.6. Homology. The (appropriately completed) equivariant homology H T Z * (Gr) of the infinite Grassmannian is a Hopf algebra dual to H * T Z (Gr) ∼ = Λ(x||a). Non-equivariantly, this can be explained by the homotopy equivalence Gr ∼ = ΩSU (∞) with a group. Restricting to a one-dimensional torus C × ⊂ T Z , the multiplication is induced by the direct sum operation on finite Grassmannians, and was studied in some detail by Knutson and Lederer [KL]. The geometry of the full multiplication on H T Z * (Gr) is still mysterious to us, and we hope to study it in the context of the affine infinite Grassmannian in the future.
We identify (see Remark 6.1) the Schubert basis of H T Z * (Gr) with Molev's dual Schur functionŝ s λ (y||a) [Mol09]. We use this to resolve (Theorem 6.11) a question posed in [KL]: to find deformations of Schur functions that have structure constants equal to the Knutson-Lederer direct sum product.
One of our main results (Theorem 6.5) is a recursive formula forŝ λ (x||a) in terms of novel homology divided difference operators, which are divided difference operators on equivariant variables, but conjugated by the equivariant Cauchy kernel. A similar formula had previously been found independently by Naruse [Nar], who was studying the homology of the infinite Lagrangian Grassmannian. Our construction is also closely related to the presentation of the equivariant homology of the affine Grassmannian given by Bezrukavnikov, Finkelberg, and Mirkovic [BFM]. We hope to return to the affine setting in the future.
We compute the ring structure of this equivariant homology ring by giving a positive Pieri rule (Theorem 6.17). Our computation of the Pieri structure constants relies on some earlier work of Lam and Shimozono [LaSh12] in the affine case, and on triple Stanley symmetric functions F w (x||a||b). The double Stanley symmetric functions F w (x||a) are recovered from F w (x||a||b) by setting b = a. The triple Stanley symmetric functions distinguish "stable" phenomena from "unstable" phenomena in the limit from the affine to the infinite setting. 1.7. Affine Schubert calculus. Our study of back stable Schubert calculus is to a large extent motivated by our study of the Schubert calculus of the affine flag variety Fl, and in particular Lee's recent definition of affine Schubert polynomials [Lee]. There is a surjection H * (Fl) → H * ( Fl n ) from the cohomology of the infinite flag variety to that of the affine flag variety of SL(n). A complete understanding of this map yields a presentation for the cohomology of the affine flag variety. Thus this project can be considered as a first step towards understanding the geometry and combinatorics of affine Schubert polynomials and their equivariant analogues.
We shall apply back stable Schubert calculus to affine Schubert calculus in future work [LLSb]. In particular, our coproduct formulae (Theorems 3.16 and 4.14) hold for equivariant Schubert classes in the affine flag variety of any semisimple group G [LLSa]. This affine coproduct formula explains certain Schubert polynomial formulae of Billey and Haiman [BH].
1.8. Other directions. Most of the results of the present work have K-theoretic analogues. We plan to address K-theory in a separate work [LLSc].
The results in this paper (for example, §8.1) suggests the study of the affine infinite flag variety Fl, an ind-variety whose torus-fixed points are the affine infinite symmetric group S Z ⋉ Q ∨ Z , where Q ∨ Z is the Z-span of root vectors e i − e j for i = j integers and e i is the standard basis of a lattice with i ∈ Z. Curiously, Schubert classes of Fl can have infinite codimension (elements of S Z ⋉ Q ∨ Z can have infinite length) and should lead to new phenomena in Schubert calculus.
Acknowledgements. We thank Anna Weigandt, Zach Hamaker, Anders Buch, and especially Allen Knutson for their comments on and inspiration for this work.
2.1.1. Permutations. Let S Z denote the subgroup of permutations of Z generated by s i for i ∈ Z where s i exchanges i and i + 1. This is the group of permutations of Z that move finitely many elements. Let S + (resp. S − ) be the subgroup of S Z generated by s 1 , s 2 , . . . (resp. s −1 , s −2 , . . .). We write S =0 = S − × S + . We have S + = n≥1 S n . Let w (n) 0 ∈ S n be the longest element. For x, y, z ∈ S Z , we write z . = xy if z = xy and ℓ(z) = ℓ(x) + ℓ(y). This notation generalizes to longer products z . = x 1 x 2 · · · x r . For w ∈ S Z , let R(w) be the set of reduced words of w. Let γ : S Z → S Z be the "shifting" automorphism γ(s i ) = s i+1 for all i ∈ Z.
For a fixed k ∈ Z, say that w ∈ S Z is k-Grassmannian if ws i > w for all i ∈ Z − {k}. We write S 0 Z for the set of 0-Grassmannian permutations. For any w ∈ S Z , let I w,+ := Z >0 ∩ w(Z ≤0 ) (2.1) The map w → (I w,+ , I w,− ) is a bijection from S 0 Z to pairs of finite sets (I + , I − ) such that I + ⊂ Z >0 , I − ⊂ Z ≤0 , and |I + | = |I − |.
2.1.2. Partitions. Let Y denote the set of partitions or Young diagrams. Throughout the paper, Young diagrams are drawn in English notation: the boxes are top left justified in the plane. For a Young diagram λ, we let λ ′ denote the conjugate (or transpose) Young diagram. The dominance order on partitions is given by There is a bijection between Y and S 0 Z , given by λ → w λ , where . .] be the polynomial ring in infinitely many positively-indexed variables and Q[x] := Q[. . . , x −1 , x 0 , x 1 , . . . ] the polynomial ring in variables indexed by integers. Define the Q-algebra automorphism γ : We have the operator identities For w ∈ S Z this allows the definition of (1) The kernel of A i is the subalgebra of s i -invariant elements.
(2) The image of A i consists of s i -invariant elements.
For w ∈ S n , the Schubert polynomial S w ∈ Q[x + ] is defined by The polynomials S w (x) are well-defined for w ∈ S n by (2.7) and (2.8).
Proof. It suffices to show that the definitions of S w (n) 0 and S w (n+1) 0 are consistent. Using w We recall the monomial expansion of S w due to Billey, Jockusch, and Stanley. Theorem 2.4 ( [BJS]). For w ∈ S + , we have The support of an indexed collection of integers (c i | i ∈ J) is the set of i ∈ J such that c i = 0. The code gives a bijection from S Z to finitely-supported sequences of nonnegative integers (. . . , c −1 , c 0 , c 1 , . . . ). It restricts to a bijection from S + to finitely-supported sequences of nonnegative integers (c 1 , c 2 , . . . ).
The following triangularity of Schubert polynomials with monomials can be clearly seen from Bergeron and Billey's rc-graph formula for Schubert polynomials [BB]. For two monomials x b and Proposition 2.5. The transition matrix between Schubert polynomials and monomials is unitriangular: S w (x) = x c(w) + reverse-lex lower terms. (2.14) Theorem 2.6. The Schubert polynomials are the unique family of polynomials {S w (x) ∈ Q[x + ] | w ∈ S + } satisfying the following conditions: Proof. For uniqueness, by induction we may assume that S ws i (x) is uniquely determined for all i such that ws i < w. Since the applications of all the A i are specified on S w , the difference of any two solutions of (2.17), being in the kernel of all A i , is S + -invariant by Lemma 2.2. But Q[x + ] S + = Q so the homogeneity assumption implies that the two solutions must be equal.
For existence, we note that the Schubert polynomials satisfy (2.15), (2.16), and (2.17) when ws i < w. When ws i > w, we have S w = A i S ws i by (2.17) applied for ws i . The element S w , being in the image of A i , is s i -invariant and therefore is in ker A i by Lemma 2.2. That is, A i S w = 0, establishing (2.17).
The basis property holds by Proposition 2.5.
Remark 2.7. All the basis theorems for Schubert polynomials and their relatives, such as Theorem 2.6, hold over Z.
Proof. Uniqueness is proved as in Theorem 2.6. For existence, let and using induction we have S v (wa; a) = (a i − a i+1 ) −1 (S vs i (wa; a) − S vs i (ws i a; a)) = 0, proving (2.22).
The basis property follows from the fact that S w (x; 0) = S w (x) are a Q-basis of Q[x + ].
2.4. Double Schubert polynomials into single. The following identity is proved in Appendix B.
2.5. Left divided differences. Let A a i be the divided difference operator acting on the a-variables. Lemma 2.11. For i > 0 and w ∈ S + , Proof. This is easily verified using Proposition 2.10.
3. Back stable Schubert polynomials 3.1. Symmetric functions in nonpositive variables. For b ∈ Z, let Λ(x ≤b ) be the Q-algebra of symmetric functions in the variables x i for i ∈ Z with i ≤ b. We write Λ = Λ(x ≤0 ) = Λ(x − ), emphasizing that our symmetric functions are in variables with nonpositive indices.
The tensor product Λ ⊗ Λ is isomorphic to the Q-algebra of formal series of bounded total degree in x − and a − which are separately symmetric in x − and a − . Under this isomorphism, we have g ⊗ h → g(x − )h(a − ). We use this alternate notation without further mention.
The Q-algebra Λ is a Hopf algebra over Q, generated as a polynomial Q-algebra by primitive elements Equivalently, for f ∈ Λ, ∆(f ) is given by plugging both x − and a − variable sets into f . The counit takes the coefficient of the constant term, or equivalently, is the Q-algebra map sending p k → 0 for all k ≥ 1. The antipode is the Q-algebra automorphism sending p k → −p k for all k ≥ 1. For a symmetric function f (x) we write f (/x) for its image under the antipode. The superization map is the Q-algebra homomorphism defined by applying the coproduct ∆ followed by applying the antipode in the second factor. Equivalently, it is the Q-algebra homomorphism sending p k → p k (x − ) − p k (a − ). In particular, f (x/a) is symmetric in x − and symmetric in a − . We use the notation f (x/a) instead of f (x − /a − ) for the sake of simplicity.
3.2. Back symmetric formal power series. Let R be the Q-algebra of formal power series f in the variables x i for i ∈ Z such that f has bounded total degree (there is an M such that all monomials in f have total degree at most M ) and the support of f is bounded above (there is an N such that the variables x i do not appear in f for i > N ). The group S Z acts on R by permuting variables.
R be the subset of back symmetric elements of R.
Proposition 3.1. We have the equality . . ] is a polynomial in the power sums p k (x ≤b ) and the variables x b+1 , x b+2 , . . . . But We emphasize that ← − R is a polynomial Q-algebra with algebraically independent generators p k for k ≥ 1 and x i for i ∈ Z. The restriction of the action of S Z from R to ← − R is given on algebra generators by The divided difference operators A i for i ∈ Z act on ← − R using the same formula as (2.5).
3.3. Back stable limit. Let γ : Given w ∈ S Z , let [p, q] ⊂ Z be an interval that contains all non-fixed points of w. Let S [p,q] w be the usual Schubert polynomial but computed using the variables x p , x p+1 , . . . , x q instead of starting with x 1 . This is the same as shifting w to start at 1 instead of p, constructing the Schubert polynomial, and then shifting variables to start at x p instead of x 1 . That is, We say that the limit of a sequence f 1 , f 2 , . . . of formal power series is equal to a formal power series f if, for each monomial M , the coefficient of M in f 1 , f 2 , . . . eventually stabilizes and equals the coefficient in f .
w called the back stable Schubert polynomial. It has the monomial expansion Moreover the back stable Schubert polynomials are the unique family { ← − S w ∈ ← − R | w ∈ S Z } of elements satisfying the following conditions: Proof. The well-definedness of the series and its monomial expansion follows by taking the limit of (2.12). Let w ∈ S Z . For i ≪ 0 we have ws i > w. By (2.17) and Lemma 2.2 ← − S w is s i -symmetric. Thus ← − S w is back symmetric. Properties (3.6), (3.7) and (3.8) hold for ← − S w by the corresponding parts of Theorem 2.6 for usual Schubert polynomials.
Proof. Let 0 < k < n be large enough such that λ is contained in the k×(n−k) rectangular partition. For such partitions the map λ → γ k (w λ ) defines a bijection to the k-Grassmannian elements of S n . It is well-known that S γ k (w λ ) = s λ (x 1 , . . . , x k ) [Ful,Chapter 10,Proposition 8].
The result follows by letting k, n → ∞.
By Propositions 3.3 and 2.5 we have ← − S w = x c(w) + reverse-lex lower terms. (3.9) Theorem 3.5. The back stable Schubert polynomials form a Q-basis of ← − R .
Proof. By (3.9) the back stable Schubert polynomials are linearly independent. For spanning, using Proposition 3.3 and applying γ n for n sufficiently large, it suffices to show that any element of Λ(x − )⊗ Q[x + ] is a Q-linear combination of finitely many back stable Schubert polynomials. This holds due to the unitriangularity (3.9) of back stable Schubert polynomials with monomials and the following facts: (1) the reverse-lex leading monomial x β in any nonzero element of Λ( (4) There are finitely many γ below β in reverse-lex order such that x γ and x β have the same degree, and satisfying · · · ≤ γ −2 ≤ γ −1 ≤ γ 0 .

Stanley symmetric functions.
There is a Q-algebra map η 0 : Q[x] → Q given by evaluation at zero: x i → 0 for all i ∈ Z. This induces a Q-algebra map 1 ⊗ η 0 : ← − R → Λ ⊗ Q Q ∼ = Λ, which we simply denote by η 0 as well.
Remark 3.6. The map η 0 "knows" the difference between x i ∈ Q[x] and the x i that appear in Λ = Λ(x − ).
For w ∈ S Z , we define the Stanley symmetric function by Recall the shifting automorphism γ : S Z → S Z from §2.1.
Proof. This holds since η 0 is a Q-algebra homomorphism and the claim is easily verified for the algebra generators of ← − R .
Theorem 3.9 (cf. [Sta]). For w ∈ S Z , we have Proof. By Lemma 3.7 we may assume that w ∈ S + . Since ws i > w for i < 0, ← − S w is s i -symmetric for i < 0, that is, . Therefore F w is obtained from ← − S w by setting x i = 0 for i ≥ 1. Making this substitution in (3.5) yields (3.11).
Remark 3.10. Up to using x − instead of x + , our definition agrees (by Theorem 3.9) with the standard definition [Sta] of Stanley symmetric function: F w (x + ) = lim n→∞ S γ n (w) (x + ).
The Edelman-Greene coefficients j w λ ∈ Z are defined by Remark 3.11. These coefficients are known to be nonnegative and have a number of combinatorial interpretations: leaves of the transition tree [LaSc85], promotion tableaux [Hai], and peelable tableaux [RS]. In particular, by [EG] j w λ is equal to the number of reduced word tableaux for w: that is, row strict and column strict tableaux of shape λ whose row-reading words are reduced words for w.
Let ω be the involutive Q-algebra automorphism of Λ defined by ω(p r ) = (−1) r−1 p r for r ≥ 1. We have ω(s λ ) = s λ ′ for λ ∈ Y. The action of ω on a homogeneous element of degree d is equal to that of the antipode times (−1) d . Let ω also denote the automorphism of S Z given by s i → s −i for all i ∈ Z.
Proof. Reversal of a reduced word gives a bijection R(w) → R(w −1 ) that sends a Coxeter-Knuth class of shape λ (see §5.8) to a Coxeter-Knuth class of shape λ ′ . The first equality follows.
Negating each entry of a reduced word gives a bijection R(w) → R(ω(w)) which sends a Coxeter-Knuth class of shape λ to a Coxeter-Knuth class of shape λ ′ . The second equality follows.
3.5. Negative Schubert polynomials. Recall S − and S =0 from §2.1. Recall the automorphism ω of S Z . It restricts to an isomorphism S − → S + . Let ω : That is, in w replace the negatively-indexed reflections with positively-indexed ones, take the usual Schubert polynomial, and then use ω to substitute nonpositively-indexed x variables for the positively-indexed ones (with signs).
Example 3.14. For u = s −3 s −2 s −1 , we have ω(u) = s 3 s 2 s 1 , S s 3 s 2 s 1 = x 3 1 , and S u = −x 3 0 . By Theorem 2.4, we have 3.6. Coproduct formula. There is a coaction ∆ : Theorem 3.16. Let w ∈ S Z . We have the coproduct formulae Proof. Equation (3.20) can be deduced from (3.21) and Proposition 3.13: We prove (3.21) by a cancellation argument. We say that a pair of integer sequences (a, b) of the same length is a compatible pair, if b is weakly increasing and a i < a i+1 =⇒ b i < b i+1 .
Let (x, y, a, b) index a monomial x b = x b 1 · · · x b ℓ on the right hand side, corresponding to the term F x S y and reduced word a = a 1 a 2 · · · a ℓ . By convention, to obtain a, we always factorize y ∈ S =0 as y = y ′ y ′′ with y ′ ∈ S − and y ′′ ∈ S + . We will provide a partial sign-reversing involution ι on the quadruples (x, y, a, b); the left-over monomials will give the left hand side.
Suppose ℓ(x) = r, ℓ(y ′ ) = s, ℓ(y ′′ ) = t, and set ℓ = r + s + t. Call an index i ∈ [1, ℓ] bad if b i > a i , and good if b i ≤ a i . It follows from the definitions that all indices i ∈ [r + s + 1, ℓ] are good, while all indices i ∈ [r + 1, r + s] are bad. Furthermore, if i ∈ [1, r] is bad, then a i < 0.
Let k be the largest bad index in [1, r], which we assume exists. We claim that s a k commutes with s a k+1 · · · s ar . To see this, observe that if a k ′ ∈ {a k − 1, a k , a k + 1} where k < k ′ ≤ r then we must have b k ′ > a k ′ , contradicting our choice of k. If s = 0, we set (3.22) ι(x, y, a, b) = (a 1 · · ·â k · · · a r |a k a r+1 · · · a t , b 1 · · ·b k · · · b r |b k b r+1 · · · b t ) = (x,ỹ,ã,b) where the vertical bar separates x from y.
and a k < a r+1 ) then we again make the definition (3.22) where now y ′ = s a k y ′ . We call this CASE A. Suppose still that s > 0. If (b k < b r+1 ) or (b k = b r+1 and a k ≥ a r+1 ) or (k does not exist) then there is a unique index j ∈ [k, r] so that ι(x, y, a, b) = (a 1 · · · a j a r+1 a j+1 · · · a r |a r+2 · · · a t , b 1 · · · b j b r+1 b j+1 · · · b r |b r+2 · · · b t ) = (x,ỹ,ã,b) has the property that (a 1 · · · a j a r+1 a j+1 · · · a r , b 1 · · · b j b r+1 b j+1 · · · b r ) is a compatible sequence. In this case, s a r+1 commutes with s a j+1 · · · s ar . We call this CASE B.
Finally, if s = 0 and k does not exist, then ι is not defined. It remains to observe that CASE A and CASE B are sent to each other via ι, which keeps x b constant and changes ℓ(y ′ ) by 1.
Let ω be the involutive Q-algebra automorphism of ← − R given by combining the maps ω on Λ from §3.4 and on Q[x] from §3.5.
Proof. This follows immediately from Theorem 3.16 and Propositions 3.12 and 3.15.
denote the linear map given by "taking the coefficient of s λ ". Then We will give an explicit description of the polynomial ν λ ( ← − S w ) in Theorem 5.11. 3.7. Back stable Schubert structure constants. For u, v, w ∈ S + , define the usual Schubert structure constants c w uv by (1) For u, v, w ∈ S + , we have c w uv = ← − c w uv .
(2) Every back stable Schubert structure constant is a usual Schubert structure constant.
Proof. Consider the Q-algebra homomorphism π + : Applying π + to Theorem 3.16 for y ∈ S Z we have (3.27) because π + kills all symmetric functions with no constant term and all negative Schubert polynomials of positive degree. Now let u, v ∈ S + . Applying π + to (3.25) and using (3.27), (1) follows.
For (2), let u, v ∈ S Z . By (3.26), we may assume that u, v ∈ S + and that the finitely many w appearing in (3.25) are also in S + . The proof is completed by applying part (1).
We derive a relation involving back stable Schubert structure constants and Stanley coefficients.
Using the algebra map η 0 several times we obtain Taking the coefficient of s λ we obtain (3.28).
The algebra Λ(x||a) is a Hopf algebra over Q[a] with primitive generators p k (x||a) for k ≥ 1. The counit is the Q[a]-algebra homomorphism ǫ : Λ(x||a) → Q[a] given by p k (x||a) → 0 for k ≥ 1. The antipode is the Q[a]-algebra homomorphism defined by p k (x||a) → −p k (x||a) for k ≥ 1.
For w ∈ S Z , we write w x (resp. w a ) for this action of w on the x-variables (resp. a-variables).

4.3.
Localization of back symmetric formal power series. Let ǫ : In other words ǫ "sets all x i to a i " including those in p k (x||a). Define for f (x, a) ∈ ← − R (x; a) and w ∈ S Z . (4.5) Recall the notation I w,± from (2.1) and (2.2).
Lemma 4.1. We have p k (x||a)| w = i∈I w,+ a k i − i∈I w,− a k i .

Back stable double Schubert polynomials.
Let γ be the Q-algebra automorphism of ← − R (x; a) which shifts all variables forward by 1 in As before, let [p, q] be an interval of integers containing all integers moved by w ∈ S Z . Define Remark 4.2. There is a double version of the monomial expansion (Theorem 2.4) of Schubert polynomials, see for example [FK]. However, the well-definedness of ← − S w (x; a) is not apparent from that expansion. In Theorem 5.13 we give a new combinatorial formula for S w (x; a) using bumpless pipedreams as a sum of products of binomials x i − a j . Theorem 5.13 is compatible with the back stable limit and yields a monomial formula (Theorem 5.2) for the back stable double Schubert polynomials.
Proof. Since length-additive factorizations are well-behaved under shifting it follows that using Proposition 2.10. Taking the limit as p → −∞ and q → ∞ we obtain (4.8).
In particular, Proof. Using (3.21) and Propositions 4.3 and 3.13 we have is the unique family of power series satisfying the following conditions: Proof. Uniqueness follows as in the proof of Theorem 3.2. Since the double Schubert polynomials are related by divided differences, the corresponding fact (4.12) also holds. For (4.11), applying the map ǫ of §4.3 to (4.9) and using (3.16), we have This is 0 automatically if w ∈ S =0 . If w ∈ S =0 \ {id} then the vanishing follows from the straightforward generalization of Lemma 2.9 to S w for w ∈ S =0 .
The basis property follows from the fact that setting The back stable double Schubert polynomials localize the same way that ordinary double Schubert polynomials do in the following sense.
Proposition 4.7. Let v, w ∈ S Z and let [p, q] ⊂ Z be an interval that contains all elements moved by v and by w.
Proof. By Corollary 4.4 we may assume that [p, q] = [1, n] for some n so that v, w ∈ S n . We are specializing x i → a w(i) for all i, and in particular x i → a i for all i ≤ 0. Under this substitution all the super Stanley functions in (4.9) vanish except those indexed by the identity. Using Proposition 2.10, we have

Double Schur functions.
We realize the double Schur functions as the Grassmannian back stable double Schubert polynomials. As such our double Schur functions are symmetric in x − . In Appendix A a precise dictionary is given which connects our conventions with the literature, which uses symmetric functions in x + .
Let γ a be the shift of all of the a-variables, that is, the Q-algebra automorphism of Λ(x||a) given by By definition γ a leaves the x variables alone. For λ ∈ Y define the double Schur function s λ (x||a) ∈ Λ(x||a) by Let s a i and A a i be the reflection and divided difference operators acting on the a-variables in both Q[a] and in p r (x||a).
The following result follows from Proposition 4.8.
, which we simply denote by η a as well.
Remark 4.10. Analogously to η 0 in Remark 3.6, the map η a substitutes x i → a i for the x i generators of Q[x] but leaves the "x i in Λ(x||a)" alone.
Proof. This follows from the definition (4.16) and Corollary 4.5.
x||a) by Proposition 4.9, since η a is the identity when restricted to Λ(x||a).

Negative double Schubert polynomials.
Let ω be the involutive Q-algebra automorphism of Q[x; a] given by ω( Proof. Equation (4.20) is straightforwardly reduced to the case that w ∈ S + , which is Proposition 2.10. Equation 4.21 follows from (4.20) by Corollary B.3.

Coproduct formula.
There is a coaction ∆ : , defined by the comultiplication on the first factor of the tensor product Λ Theorem 4.14. Let w ∈ S Z . We have the coproduct formulae Proof. We first deduce (4.22) from (4.23). Using Corollary 4.5, Proposition 4.11 and Lemma 2.9 we have For (4.23), using Propositions 4.11 and 4.13 we have Corollary 4.15. Let w ∈ S Z . Then ← − R (x; a) by acting as η a on the second factor. The result follows from (4.22).
Alternatively, (4.28) follows from the uniqueness of the Schubert basis as defined by localizations.
Proof. By Theorem 4.6, we have ← − S w (a; a) = 0 if w = id and ← − S id (a; a) = 1. The result follows by localizing both sides of (4.31) at id.
is a positive integer polynomial in the linear forms a i − a j where i ≺ j under the total ordering of Z given by Theorem 4.20 will be proven in Section 8.8. Define the coproduct structure constantsĉ Proof. This holds by taking the coefficient of s ν (x||a) ⊗ s µ (x||a) in (4.24).
Remark 4.22. In Theorem 6.17 we give a formula forĉ λ µν (a) which is positive in the sense of Theorem 4.20 in the special case that µ or ν is a hook.
Let d(λ) be the Durfee square of λ ∈ Y, the maximum index d such that λ d ≥ d. The following change of basis coefficients between the double and super Schur bases were previously computed in [ORV] expressed as a determinantal formula and in [Mol09] by a tableau formula. We give them as (signed) Schubert polynomials.
= uvz, arguing as in the proof of Corollary 4.16, we first have z ∈ S 0 Z ∩ S =0 = {id}. Next we deduce that v = w µ for some µ ∈ Y such that µ ⊂ λ. Thus u = w λ/µ . The condition w λ/µ ∈ S =0 holds if and only if the skew shape λ/µ contains no boxes on the main diagonal, that is, d(λ) = d(µ). This proves (4.33).

Bumpless pipedreams
We shall consider various versions of bumpless pipedreams. These are tilings of some region in the plane by the tiles: empty, NW elbow, SE elbow, horizontal line, crossing, and vertical line.
We shall use matrix coordinates for unit squares in the plane. Thus row coordinates increase from top to bottom, column coordinates increase from left to right, and (i, j) indicates the square in row i and column j.
5.1. S Z -bumpless pipedreams. Let w ∈ S Z . A w-bumpless pipedream is a bumpless pipedream covering the whole plane, satisfying the following conditions: (1) there is a bijective labeling of pipes by integers; (2) the pipe labeled i eventually heads south in column i and heads east in row w −1 (i); (3) two pipes cannot cross more than once; (4) for all N ≫ 0 and all N ≪ 0, the pipe labeled N travels north from (∞, N ) to the square (N, N ) where it turns east and travels towards (N, ∞). Because of condition (2), every pipe has to make at least one turn. We call pipe i standard if it makes exactly one turn and this turn is at the diagonal square (i, i). By (4), all but finitely many pipes are standard. We often omit standard pipes from our drawings of pipedreams. The weight wt(P ) := (x i − a j ) of a pipedream P is the product of x i − a j over all empty tiles (i, j).
Example 5.1. Let w = s 3 s 0 s 1 . In one line notation, w(−2, −1, 0, 1, 2, 3, 4) = (−2, −1, 1, 2, 0, 4, 3) and the rest are fixed points. The following is a w-bumpless pipedream, where we have only drawn the region {(i, j) | i, j ∈ [−2, 4]}. In the left picture, the empty tiles have been indicated, as have the row and column numbers. The label of a pipe is the column number that its south end is attached to. In the right picture, we have indicated the labels of the pipes instead of the row numbers. The one-line notation of w can then be read off the east border.
where the sum is over all w-bumpless pipedreams.

Drooping and the Rothe pipedream.
A w-bumpless pipedream is uniquely determined by the location of the two kinds of elbow tiles. Each pipe has to turn at least once. There is a unique w-bumpless pipedream such that for all i, pipe i turns right from south to east in the square (w −1 (i), i). We call this the Rothe pipedream D(w) of w. The empty tiles of the Rothe pipedream form what is commonly known as the Rothe diagram of w.
Let P be a w-bumpless pipedream. A droop is a local move that swaps a SE elbow e with an empty tile t, when the SE elbow lies strictly to the northwest of the empty tile. Let R be the rectangle with northwest corner e and southeast corner t and let p be the pipe passing through e. After the droop, the pipe p travels along the southmost row and eastmost column of R; a NW elbow occupies the square that used to be empty while the square that contained a SE elbow turns becomes empty. The droop is allowed only if: (1) the westmost column and northmost row of R contains p, (2) the rectangle R contains only one elbow which is at e, and (3) after the droop we obtain a bumpless pipedream P ′ . Pipes p ′ = p do not move in a droop. We denote a droop by P ց P ′ . A Rothe bumpless pipedream is shown followed by a sequence of two droops: Proof. Let P ′ be a w-bumpless pipedream and e be a NW elbow that is northwestmost among NW elbows in P ′ . Let p be the pipe passing through e. Then p passes through SE elbows f (resp. f ′ ) in the same row (resp. column) as e. Let R be the rectangle bounded by e, f, f ′ with northwest corner t. It is easy to see that t must be an empty tile and R does not contain any other elbows. Thus there is a droop P ց P ′ which occurs in the rectangle R, and P has strictly fewer NW elbows than P ′ . Repeating, we we eventually arrive at the Rothe pipedream.
Corollary 5.4. The number of empty tiles in a w-bumpless pipedream is equal to ℓ(w).

5.3.
Halfplane crossless pipedreams. For λ ∈ Y, a λ-halfplane pipedream is a bumpless pipedream in the upper halfplane H = Z ≤0 × Z such that the crossing tile is not used, and (1) there are (unlabeled) pipes entering from the southern boundary in the columns indexed by I ⊂ Z; (2) setting (I + , I − ) = (I ∩ Z >0 , Z >0 \ I), we have I ± = I w λ ,± (see (2.1), (2.2), and (2.3)); (3) the i-th eastmost pipe entering the south heads off to the east in row 1 − i. (Equivalently, for every row i ∈ Z ≤0 , there is some pipe heading towards (i, ∞).) As before, the weight of a λ-halfplane pipedream is wt(P ) = (x i − a j ), where the product is over all empty tiles (i, j) in the halfplane H.
A Z ≤0 -semistandard Young tableau of shape λ is a filling of the Young diagram λ (in English notation) with the integers 0, −1, −2, . . . such that rows are weakly increasing and columns are strictly increasing. The weight wt(T ) of a Z ≤0 -SSYT is the product wt(T ) = (i,j)∈T (x T (i,j) − a T (i,j)+c(i,j) ) where c(i, j) = j − i is the content of the square (i, j) in row i and column j.
Corollary 5.7. Let λ be a partition. Then s λ (x||a) = T wt(T ) where the sum is over all Z ≤0 -SSYT of shape λ. 5.4. Rectangular S n -bumpless pipedreams. Let w ∈ S n . A w-rectangular bumpless pipedream is a bumpless pipedream in the n × 2n rectangular region The pipes are labeled 1 − n, 2 − n, . . . , 0, 1, . . . , n, entering the south boundary from left to right. The positively labeled pipes exit the east boundary: pipe i exits in row i. The nonpositively labeled pipes exit the north boundary. Two pipes intersect at most once, and a nonpositively labeled pipe cannot intersect any other pipe. As before, the weight of a rectangular S n -bumpless pipedream P is given by wt(P ) = (x i − a j ), with the product over all empty tiles (i, j).
Lemma 5.8. Let w ∈ S n . Suppose P is a S Z -bumpless pipedream for w (considered an element of S Z . Then the region inside the rectangle R n is a S n -rectangular bumpless pipedream for w. We also associate a partition λ(P ) to a S n -rectangular bumpless pipedream: it is obtained by reading the North boundary edges from right to left, to then obtain the boundary of a partition inside a n × n box, where empty edges (marked e below) correspond to steps to the left, and edges with a pipe exiting (marked x below) correspond to downward steps. See Figure 1.
Lemma 5.9. Let w ∈ S n and P be a w-bumpless pipedream. We have ℓ(w) = |λ(P )| + deg(wt(P )).  Example 5.10. Let w = 2143. In Figure 2 is a complete list of all w-bumpless pipedreams. Nonpositively labeled pipes are drawn in red.
Theorem 5.11. Let w ∈ S n . Then we have ← − S w (x; a) = P wt(P )s λ(P ) (x||a) where the sum is over all w-rectangular bumpless pipedreams.
Theorem 5.11 is proved in Section 11.3.
Corollary 5.12. Let w ∈ S n . Then F w (x||a) = P η a (wt(P ))s λ(P ) (x||a) where the sum is over all w-rectangular bumpless pipedreams.  (1) 5.5. Square S n -bumpless pipedreams. Let w ∈ S n . A w-square bumpless pipedream is a bumpless pipedream in the n × n square region The pipes are labeled 1, . . . , n, entering the south boundary from left to right. The pipes exit the east boundary: pipe i exits in row i. Two pipes intersect at most once. As before, the weight of a square S n -bumpless pipedream P is given by wt(P ) = (x i − a j ), with the product over all empty tiles (i, j). In Example 5.10, if we erase the left half and all red pipes, we obtain a square 2143-bumpless pipedream.
Theorem 5.13. Let w ∈ S n . Then we have S w (x; a) = P wt(P ) where the sum is over all w-square bumpless pipedreams.
Proof. By Theorem 4.14 and Lemma 4.18, when summed over w-rectangular bumpless pipedreams P satisfying λ(P ) = ∅. The condition λ(P ) = ∅ is equivalent to all nonpositively labeled pipes in P being completely vertical. In particular, the nonpositively labeled pipes stay within the left n × n square of P . Such pipedreams are in weightpreserving bijection with w-square bumpless pipedreams.
Proof of Theorem 5.2. The special role of the row and column indexed 0 is arbitrary. In Theorem 5.13, we could obtain a formula for the double Schubert polynomial S [p,n] w (x; a) (with variables x p , x p+1 , . . . , x n and a p , a p+1 , . . . , a n ) if we worked with square w-bumpless pipedreams in rows and columns indexed by p, p + 1, . . . , n. We note that such bumpless pipedreams are back stable: there is a natural weight-preserving injection sending such a pipedream for S [p,n] w (x; a) to a pipedream for S The union of all such square w-pipedreams are exactly the w-bumpless pipedreams of Theorem 5.2. Taking p → −∞, Theorem 5.2 thus follows from the definition of back stable double Schubert polynomial.
Proof of Theorem 5.6. We apply Theorem 5.2 to w = w λ . We have w λ (1) < w λ (2) < · · · and w λ (0) > w λ (−1) > · · · . It follows that in a w λ -bumpless pipedream: (1) there are no crossings in rows indexed by nonpositive integers; (2) there are no empty tiles in rows indexed by positive integers. Thus the lower half of a w λ -bumpless pipedream P is completely determined by λ, and the upper half is a λ-halfplane pipedream. 5.6. EG pipedreams. Let w ∈ S n . Let P be a w-square bumpless pipedream. We call P a w-EG pipedream if all the empty tiles are in the northeast corner, where they form a partition shape λ = λ(P ), called the shape of P . See Figure 3.
Theorem 5.14. The Stanley coefficient j w λ = j w λ (0) is equal to the number of w-EG pipedreams P satisfying λ(P ) = λ.
Proof. Specialzing a i = 0 for all i in Corollary 5.12, we obtain F w = P s λ(P ) where the sum is over all w-rectangular bumpless pipedreams P with no empty tiles. In particular, the positively labeled pipes in the right n × n square of P forms a w-EG-pipedream. The nonpositively labeled pipes in P have to fill up all the remaining tiles, and since they cannot intersect, there is a unique way to do so. Thus there is a bijection between w-rectangular bumpless pipedreams and w-EGpipedreams. Finally, one verifies from the definitions that λ(P ) is defined consistently for the two kinds of pipedreams.
An empty tile T in a bumpless pipedream D is called a floating tile if there exists a pipe that is northwest of T . A bumpless pipedream D is called near EG if it has a single floating tile. 5.7. Column moves. We define column moves that modifies a bumpless pipdream in two adjacent columns. Sometimes a column move is a droop.

→ → → →
Only one of the pipes (the active pipe) is drawn in these pictures. For the move to be allowed, the southeastmost tile must be an empty tile (before the move), and it must be the only empty tile. Thus the move takes the empty tile from the southeastmost position to the northwestmost position. There are usually other pipes (indicated in black) in the move, and the "kinks are shifted left" if necessary: → A column move is a droop if no kinks are present, and in addition, the pipe exits south in the left column and exits east in the right column. We write D → D ′ if two bumpless pipedreams are related by a column move. We say that D ′ is obtained from D by a downwards column move.
Lemma 5.15. Let D be a bumpless pipedream that is not an-EG pipedream. Then there is a downwards column move D → D ′ .
Proof. Let E be any northwestmost floating tile in D. Then the tile W immediately to west of E is nonempty, and must either be a NW elbow or a vertical line. Call this pipe p. Then p travels up from W a number of tiles and turns towards the east at a tile T . We may perform a column move in the rectangle with corners T and E.
Lemma 5.16. Let D be a a near EG-pipedream. Then there is a unique sequence of moves D → D ′ → D ′′ → · · · → D * where D * is a EG-pipedream.
Write r(D) = D * for the EG-pipedream of Lemma 5.16.
Remark 5.17. We can define an equivalence relation on bumpless pipedreams using column moves. We caution the reader that multiple EG-pipedreams can belong to a single such equivalence class.
5.8. Insertion. Let D be an w-EG-pipedream and i ∈ [1, n − 1] be such that s i w > w, or equivalently, the pipes labeled i and i + 1 do not cross in D. Let D ′ be the bumpless diagram obtained from D by swapping the pipes i and i + 1 in columns i and i + 1. Namely, if in D the first turn of pipe i (resp. i + 1) is in row a i (resp. a i+1 > a i ), then in D ′ the first turn of pipe i (resp. i + 1) is in row a i+1 (resp. a i ). Other pipes that cross pipe i in column i have their "kinks shifted left" in D ′ . In the following illustration, i ′ = i + 1.
The northwestmost tile in the shown rectangle is always an empty tile in D ′ . Thus D ′ is either a EG-tableau or a near EG-tableau. Note that there are two possibilities for the northeastmost tile in the shown rectangles. We define the insertion D ← i to be the EG-pipedream given by (Note that D ← i is not defined if the pipes i and i + 1 cross.) Let D 0 be the unique EG-pipedream for the identity permutation. Let i = i 1 i 2 · · · i ℓ be a reduced word. Then define (P, Q) = (P (i), Q(i)) by Note that Q(i) is a saturated chain of partitions, and is thus equivalent to a standard Young tableau of shape λ(P (i)). We recall the Coxeter-Knuth equivalence relation on the set R(w) of reduced words for w. It is generated by the elementary relations · · · ikj · · · ∼ · · · kij · · · for i < j < k · · · ikj · · · ∼ · · · jki · · · for i < j < k · · · i(i + 1)i · · · ∼ · · · (i + 1)i(i + 1) · · · .
Edelman-Greene insertion provides a bijection C → T (C) between Coxeter-Knuth equivalence classes C ⊂ R(w) and reduced word tableaux T for w (see [EG] and Remark 3.11). Bumpless pipedreams encode a new version of Edelman-Greene insertion.
Theorem 5.18. The map i → (P (i), Q(i)) induces a bijection between reduced words for S n and pairs consisting of an EG-pipedream and a standard Young tableau of the same shape. For a fixed EG-pipedream D, the set C D := {i | P (i) = D} is a single Coxeter-Knuth equivalence class. The shape of the reduced word tableau T (C D ) is λ(D).
Problem 5.19. Find a direct shape-preserving bijection between EG-pipedreams for w and reduced word tableaux for w.
Remark 5.20. There is a transpose analogue of column moves called row moves. We can also define insertion into EG-pipedreams using row moves. Theorem 5.18 holds with (usual) Edelman-Greene insertion replaced by Edelman-Greene column insertion.
The insertion path of the insertion D ← i is the collection of positions that the empty tile travels through in the calculation of r(D ′ ). An insertion path consists of a number of boxes, one in each of an interval of columns. Two insertion paths are compared by comparing respective boxes in the same column.
The following key result is immediate from the definition of column moves.
Lemma 5.21. The pair (D, i) can be recovered uniquely from the pair (D ← i, final box in the insertion path).
(1) Then the insertion path of D ← i is strictly below the insertion path of (D ← i) ← j.
(2) Then the insertion path of D ← j is strictly above the insertion path of (D ← j) ← i.
Proof. We show claim (1); claim (2) is similar. Let the insertion path of Let the insertion path of (D ← i) ← j be the boxes c j , c j−1 , . . . , c t where c k is in column k. Consider the calculation of c i : the lowest elbow in column i − 1 of D 1 is at the same height as b i . Thus c i−1 must at the same height or above b i . It follows that c i is above b i , and indeed it must be at least as high as b i+1 because there are no elbows in column i above b i and below the row of b i+1 . The claim (1) follows by repeating this argument.
Recall that the descent set Des(T ) of a standard Young tableaux T is the set of letters j such that j + 1 in a lower row than j in T . The descent set Des(i) of a word i = i 1 · · · i ℓ is the of indices j such that i j > i j+1 .
Corollary 5.23. For a reduced word i, we have Des(i) = Des(Q(i)).
Lemma 5.24. Let D be a EG-pipedream and suppose i < j < k. Then (when the EG-pipedreams are defined), Proof. We prove claim (1); claim (2) is similar. By Lemma 5.22, the insertion path for D ← j is above that of (D ← j) ← i. In particular, the two EG-pipedreams (D ← j) ← i and D ← i differ only in tiles that are below the insertion path of j. On the other hand, the insertion path for (D ← j) ← k is above that of D ← j, and thus does not see the part the pipedream below the insertion path of j. The desired equality follows.
Lemma 5.25. Let D be a EG-pipedream and suppose i, i + 1 ∈ [1, n − 1]. When the EG-pipedreams are defined, we have Proof. For the insertions to be defined, the pipes i, i + 1, and i + 2 in D do not intersect. Let h i , h i+1 , h i+2 be the heights of the boxes containing the first right elbow for the pipes i, i + 1, and i + 2 respectively. Then h i is strictly above h i+1 , which is strictly above h i+2 . Let us first consider D 1 = ((D ← i) ← i + 1) ← i. To calculate (D ← i) we will first create an empty tile in the box (i, h i ) in column i. Instead of moving this empty tile to the northwest immediately, let us keep it where it is, and consider the insertion of i + 1. This creates an empty tile in the box (i + 1, h i ). Finally, the second insertion of i creates an empty tile in (i, h i+1 ). Call the resulting bumpless diagram D ′ 1 . Checking the definitions, we see that D 1 is obtained from D ′ 1 by performing column moves on the three empty tiles, as long as we move the empty tiles in order. Now consider D 2 = ((D ← i + 1) ← i) ← i + 1. To calculate (D ← i + 1) we will first create an empty tile in the box (i + 1, h i+1 ) in column i. Applying a single downward move to this empty tile, we see that it will end up in box (i, h i ). At this point the first right elbow in column i will be at height h i+1 . Now we consider the insertion of i, which creates an empty tile at position (i, h i+1 ). Finally, the second insertion of i + 1 creates an empty tile in (i + 1, h i ). The resulting bumpless diagram is identical to D ′ 1 . To obtain D 2 , we perform column moves on the three empty tiles in the correct order.
The difference between the calculation of D 1 and D 2 is that the order of applying column moves to the empty tiles in positions (i, h i+1 ) and (i + 1, h i ) are swapped. We claim that the resulting EG-diagrams D 1 and D 2 are nevertheless identical. This is because the path of the tile at (i, h i+1 ) (resp. (i + 1, h i )) stays below (resp. above) that of the tile at (i, h i ). Thus the corresponding column moves commute, as in the proof of Lemma 5.24.
Proof of Theorem 5.18. Bijectivity is straightforward from the constructions: by Lemma 5.21, the map i → (P (i), Q(i)) is injective, and applying this reverse map to pairs (P (i), Q(i)) shows that the map is surjective.
By Lemma 5.25 and 5.24, Coxeter-Knuth equivalent reduced words have the same insertion EGpipedream. Thus the set {i | P (i) = D} is a union of Coxeter-Knuth equivalence classes. That it is a single Coxeter-Knuth equivalence class can be deduced from Theorem 5.14. Alternatively, the same claim can be deduced from the reversed versions of Lemmas 5.22, 5.25, and 5.24.
Let SYT(λ) denote the set of standard Young tableaux of shape λ. Then the collection {Des(S) | S ∈ SYT(λ)} of descent sets uniquely determines λ. (For example, this collection encodes the expansion of the Schur function s λ in terms of fundamental quasisymmetric functions, and the assignment λ → s λ is injective.) Let sh(T ) denote the shape of a Young tableau T . Then for a Coxeter-Knuth equivalence class C, the equality of multisets {Des(i) | i ∈ C} = {Des(S) | S ∈ SYT(sh(T (C)))} is known to hold for Edelman-Greene insertion. The last claim then follows from Corollary 5.23.
Define the Cauchy kernel This Proposition 6.2. For µ ∈ Y, we havê 6.2. Homology divided difference operators. Since α 0 = a 0 − a 1 , we use expressions such as Remark 6.3. The expression Ω[(a 1 − a 0 )y] should be viewed as the action of the translation element for the weight θ = a 1 − a 0 in a large rank limit of the affine type A root system.
For i ∈ Z, define the operatorss It is clear that these operators, being conjugate to the operators s a i and A a i respectively, satisfy the type A braid relations. Thus δ w = δ i 1 · · · δ i ℓ makes sense for any reduced decomposition w = s i 1 · · · s i ℓ ∈ S Z .
Since Ω[(a − x)y] is s a i invariant for i = 0 we havẽ s a i = s i for i = 0 (6.5) The last two follow by conjugating the relations s a i A a i = A a i and The diagonal index of a box in row i and column j is by definition j − i. For λ ∈ Y and d ∈ Z let λ + d denote the partition obtained by adding a corner to λ in the d-th diagonal if such a corner exists. Define λ − d similarly for removal of the corner in diagonal d if it exists. By convention, if a symmetric function is indexed by λ ± d and the relevant cell in diagonal d does not exist then the expression is interpreted as 0. In particular, by Proposition 4.8 we have A a i s λ (x||a) = −s λ−i (x||a) for all λ ∈ Y and i ∈ Z. (6.11) Proposition 6.4. For all µ ∈ Y and i ∈ Z, we have δ i (ŝ µ (y||a)) =ŝ µ+i (y||a). (6.12) Proof. Using (6.10), we have 0 = ΩA a i (1) = δ i (Ω) = δ i ( λ s λ (x||a)ŝ λ (y||a)) = λ (s λ (x||a)δ i (ŝ λ (y||a)) − s λ−i (x||a)s a i (ŝ λ (y||a)).
It follows that the dual Schurs can be created by applying the homology divided difference operators to 1.
Example 6.6. We haveŝ 1 (y||a) = p,q≥0 (−a 0 ) q a p 1 s p+1,1 q (y) = (α a 0 ) −1 (1 − Ω[−α 0 y]). Remark 6.7. This construction can also be adapted to compute the homology Schubert basis for the affine Grassmannian of G = SL k+1 , equivariant with respect to the maximal torus T of G. The resulting basis is the k-double-Schur functions of [LaSh13]. A k-double Schur function consists of a k-Schur function in its lowest degree and typically has infinitely many terms of higher degree with equivariant coefficients.

δ-dual Schurs represent Knutson-Lederer classes. Knutson and Lederer [KL] define a ring R H S that is a one-parameter deformation the symmetric functions Λ. Namely, R H S is a free
The multiplication in R H S is defined as follows. Let : Gr(a, a + b) × Gr(c, c + d) → Gr(a + c, a + b + c + d) be the direct sum map (V, W ) → V ⊕ W . Let the circle S 1 act on each C a+b by acting with weight 1 on the first b coordinates and weight 0 on the last a coordinates. This induces an action of S 1 on Gr(a, a + b). In R H S , we have where the RHS is the class in H S 1 * (Gr(a + c, a + b + c + d)) of the direct sum (X λ , X µ ) of two opposite Schubert varieties. Here a, b, c, d are chosen so that λ ⊆ a × b and µ ⊆ c × d.
The following result answers a question implicitly posed in [KL].
Theorem 6.11. There is an isomorphism of Q[δ]-algebras Proof. It suffices to show that the structure constants d ν λµ (δ) of R H S are obtained from the coproduct structure constants of the Hopf algebra Λ(x||a) after specializing a i = 0 for i ≤ 0 and a i = δ for i > 0.
For simplicity, we assume that a = b and c = d in the following calculation. Let us think of C a+c+c+a as spanned by e a+c , e a+c−1 , . . . , e 1 , e 0 , e −1 , . . . , e 1−a−c , with a natural action of T = (C × ) a+c+c+a . We identify H * T (pt) = Q[a a+c , a a+c−1 , . . . , a 1−a−c ]. We thus have actions of T on Gr(c, c + c) (the c-dimensional subspaces of span(e c , e c−1 , . . . , e 1−c )) and on Gr(a, a + a) (the adimensional subspaces of span(e a+c , e a+c−1 , . . . , e c+1 , e −c , e −c−1 , e 1−a−c )). Finally, we have a Taction on Gr(a + c, a + c + c + a) and the direct sum map is T -equivariant, so we obtain a map of H * T (pt)-modules (6.14) H T * (Gr(a + c, a + c + c + a)) → H * T (Gr(a, a + a)) ⊗ H * T (Gr(c, c + c)). Since a T -equivariant cohomology class of any of these Grassmannians is determined by its value at T -fixed points, the map (6.14) is completely determined by the direct sum map applied to T -fixed points.
The For Gr(a, a + a) we consider T -fixed points as pairs (K − , K + ) with K − ⊆ [1 − a − c, −c] and K + ⊆ [c+1, a+c]. Then the direct sum map induces the map ((K − , K + ), (J − , J + )) → (J − ∪K − , J + ∪K + ). By Proposition 7.15, this agrees with the coproduct of Λ(x||a) in terms of localization. (Note that in this work we do not give a geometric explanation of the coproduct of Λ(x||a) similar to the direct sum map, which is not equivariant with respect to the natural infinite-dimensional torus.) By Proposition 11.1, the double Schur functions s λ (x||a) can be identified with the opposite Schubert class [X λ ] in equivariant cohomology H * T (Gr(a, a + b)). It follows that the structure constants of (6.14) with respect to the opposite Schubert classes [X λ ] coincide with the coproduct structure constants (4.32) of the double Schur functions. Specializing a i = 0 for i ≤ 0 and a i = δ for i > 0 gives the desired conclusion.
Remark 6.12. Knutson and Lederer [KL] also define a K-theoretic analogue, and a result similar to Theorem 6.11 holds.
6.4. Homology equivariant Monk's rule. A vertical strip is a skew shape that contains at most one box per row. A horizontal strip is a skew shape that contains at most one box per column. A ribbon R = λ/µ is a (edgewise) connected skew shape not containing any 2 × 2 square. A skew shape λ/µ is called thin if its connected components are ribbons. We write c(λ/µ) for the number of connected components of a thin skew shape. Lemma 6.13. Let R = λ/µ be a nonempty ribbon. Then there exists exactly two shapes such that λ/ρ is a vertical strip and ρ/µ is a horizontal strip.
Proof. The northeast most square of R can belong to either λ/ρ or ρ/µ. For all other boxes b ∈ R, either R contains the square directly north of b in which case b ∈ λ/ρ or R contains the square directly east of b in which case b ∈ ρ/µ. Suppose λ/µ is a skew shape. A Λ-decomposition of λ/µ is a pair D = (λ/ρ, ρ/µ) consisting of a vertical strip and a horizontal strip. If λ/µ has a Λ-decomposition then it must be thin. In this case, it follows from Lemma 6.13 that λ/µ has exactly 2 c(λ/µ) Λ-decompositions.
The weight of a Λ-decomposition D = (λ/ρ, ρ/µ) is the product which can be 0. If D = (λ/ρ, ρ/µ) is a Λ-decomposition, let D − be obtained from D by removing the northeast most square of λ/µ from whichever of λ/ρ or ρ/µ that contains it.
Theorem 6.14. Let µ ∈ Y. We havê where the inner sum is over all distinct D − that can be obtained from some nonempty Λ-decomposition D = (λ/ρ, ρ/µ) with outer shape λ.
The proof of Theorem 6.14 will be given in Section 9.6. In the non-equivariant case with a i = 0, Theorem 6.14 reduces to the usual one-box Pieri rule: when λ/µ is a single box, there are two possible choices of D, but D − will always be empty and wt(D − ) = 1.
(The number of such T is equal to the number of semistandard Young tableaux for a single row of size |ρ/µ| − q using the numbers 1 through q + 1.) Define the weight of a q-horizontal filling T by where the product is over all boxes (i, j) such that either T (i, j) = 1 or (i, j) is not the leftmost occurrence of T (i, j) in T . Thus wt(T ) ∈ Q[a] has degree equal to |ρ/µ| − q (or is 0 if q > |ρ/µ|). Similarly, a p-vertical filling of λ/ρ is a filling T of a vertical strip λ/ρ with integers 0, −1, . . . , −p so that the numbers are weakly decreasing from top to bottom regardless of column, and every number from −1 to −p is used. Define the weight of a p-vertical filling T ′ by where the product is over all boxes (i, j) such that either T (i, j) = 0 or (i, j) is not the topmost occurrence of T (i, j) in T . A (p, q)-filling of a Λ-decomposition (λ/ρ, ρ/µ) is a pair (T ′ , T ) consisting of a p-vertical filling T ′ of λ/ρ and a q-horizontal filling T of ρ/µ. The (p, q)-weight of a Λ-decomposition D = (λ/ρ, ρ/µ) to be summed over all (p, q)-fillings (T ′ , T ) of D. We note that wt 0,0 (D) is the weight wt(D) from (6.15). Also note that if p > |λ/ρ| or q > |ρ/µ| then wt p,q (D) = 0.
The following result gives a rule for multiplication by a hook-shaped dual Schur function.
Theorem 6.17. Let µ ∈ Y and p, q ≥ 0. We havê where the inner sum is over all distinct D − that can be obtained from some nonempty Λ-decomposition D = (λ/ρ, ρ/µ) with outer shape λ.
The proof of Theorem 6.17 will be given in Section 9.7.
Remark 6.18. Suppose we forget equivariance by setting a i = 0 for all i. Let D be a nonempty Λ-decomposition with outer shape λ appearing in (6.18). Thenĉ λ µ,(q+1,1 p ) (0) is the Littlewood-Richardson coefficient, the coefficient of s λ in the product s µ s (q+1,1 p ) . The latter is the number of standard tableaux of shape λ/µ such that 1, 2, . . . , q + 1 go strictly east and weakly north, and the numbers q + 1, q + 2, . . . , q + p + 1 go strictly south and weakly west [RW]. By Theorem 6.17, in order to contribute to the sum, wt p,q (D − ) must be degree 0. This restricts the sum over (p, q)fillings (T ′ , T ) of D − such that wt p (T ′ ) = 1 = wt q (T ). For each D − there is a unique filling: in T ′ the numbers −p, . . . , −2, −1 are used once each and go strictly south and weakly west, while in T the numbers 2, 3, . . . , q + 1 are used once each and go strictly east and weakly north. These (T ′ , T ) biject with the above standard tableaux: 1 appears in the northeastmost box of D and −p, . . . , −2, −1 are replaced by q + 2, q + 3, . . . , q + p + 1. Thus the nonequivariant specialization of Theorem 6.17 agrees with the Littlewood-Richardson rule.
Remark 6.19. By Proposition 4.21,ĉ λ µν (a) = j w λ/µ ν (a). Theorem 6.17 expresses these polynomials positively in the sense of Theorem 4.20 when one of µ or ν is a hook. This should be compared with [Mol09,§4] in which a combinatorial formula is given for allĉ λ µν (a). This formula does not exhibit the positivity of Theorem 4.20.
Finally, let us consider λ = (3, 1). The box (1, 3) is ignored. The box (1, 2) must be in the horizontal strip of D − while the box (2, 1) must be in the vertical strip of D − . There is a unique filling with (1, 0)-weight (a 1 − a 2 ) which is the coefficient ofŝ 31 .

Localization
In §10 we will recall the definition of an infinite flag ind-variety Fl Z and an infinite Grassmannian Gr, both of which afford the action of an infinite torus T Z . Localization [KK, CS, GKM] provides algebraic (GKM) The (infinite) nilHecke algebra A is by definition the Q-subalgebra of Q(a)[S Z ] generated by Q[a] and the A i . We have the commutation relation Viewing by the expansion of Weyl group elements into the basis A v of A: Example 7.1. Using (7.2) we have Therefore e s 2 s 1 id = 1, e s 2 s 1 s 1 = a 2 − a 1 , e s 2 s 1 s 2 = a 3 − a 1 , e s 2 s 1 s 2 s 1 = (a 3 − a 2 )(a 3 − a 1 ), and e s 2 s 1 v = 0 for other v ∈ S Z .
Proposition 7.2. The elements e v w are uniquely defined by the initial conditions e v id = δ v,id for all v ∈ S Z (7.4) and either the following: (a) For all w ∈ S Z and i ∈ Z such that ws i < w, Let a = s i 1 s i 2 · · · s i ℓ be a reduced word in S Z . For each letter s i j in a, we define a root Proposition 7.3 ( [AJS, Bil]). Let w, v ∈ S Z and a be a reduced word for w. Then we have the closed formula summed over all subwords of a that are reduced words for v.

GKM rings for infinite flags and infinite Grassmannian. Let Fun(S Z , Q[a]) be the Q[a]algebra of all functions
Example 7.4. For p ∈ Q[a], define L p ∈ Fun(S Z , Q[a]) by L p | w = w(p). Then L p ∈ Ψ ′ . If p is homogeneous of degree one then L p is an equivariant line bundle class.
The methods of [KK] are easily adapted to prove the following.
Proposition 7.6. [KK] There exists a unique family of elements {ξ v | v ∈ S Z } ⊂ Ψ ′ (the equivariant Schubert basis) such that We define Ψ := w∈S Z Q[a]ξ w Z , which is the Q[a]-submodule of Ψ ′ with basis ξ w Z . It follows from Proposition 7.12 below that Ψ is a Q[a]-subalgebra of Ψ ′ . Let Recall the bijection λ → w λ ((2.3)).
The GKM ring Ψ affords two actions of A which commute.
(1) There is an action • of A on Ψ defined by (7.14) where L p is defined in Example 7.4. It satisfies (7.15) (2) There is an action * of A on Ψ is given by (7.17) In particular the action of S Z on Ψ by * is by conjugation: Proof. We identify any function ψ ∈ Fun(S Z , Q(a)) with the left Q(a)-module homomorphism ψ ∈ Hom Q(a) (Q(a)[S Z ], Q(a)) by extension by left Q(a)-linearity.
For (1), there is an action of Q(a)[S Z ] on Fun(S Z , Q(a)) defined by For b = u ∈ S Z , we obtain (7.15). For a = A i and a = p, we have which agrees with (7.13) and (7.14). To see that • restricts to an action of A on Ψ ′ , let ψ ∈ Ψ ′ . Note that if α = w(α i ) then ws i = s α w so that α divides ψ| w − ψ| sαw = ψ| w − ψ| ws i and For (2), again working over Q(a) the * -action is defined by (7.17) and (7.18). To show these define an action of Q(a)[S Z ] one must verify that the actions of p ∈ Q(a) and u ∈ S Z have the proper commutation relation: To check that * restricts to an action of A on Ψ ′ , let ψ ∈ Ψ ′ . Note that for any p ∈ Q[a], s i (p) − p = α i g for some g ∈ Q[a] (namely, g = −A i (p). Then ψ| s i w − ψ| w = α i h for some h ∈ Q[a]. We have For (3) it is straightforward to check over Q(a) that the operators p• and A i • commute with the operators q * and A j * .
The ring Fun(S Z , Q[a]) has a product structure given by pointwise multiplication: (ψφ)| w = ψ| w φ| w for ψ, φ ∈ Fun(S Z , Q[a]). It is easy to see that this induces a structure of a Q[a]-algebra on Ψ ′ .
The nilHecke algebra A has a comultiplication map ∆ : A → A ⊗ Q[a] A given by and extending by linearity over Q(a). One can show that (7.21) is equivalent to We caution that A is not a Hopf algebra. Define a pairing · , · : Fun(S Z , Q(a)) ⊗ Q(a) Q(a)[S Z ] → Q(a) by where a w ∈ Q(a).
(2) The multiplication of Ψ ′ is dual to the comultiplication ∆ of A.

Isomorphism with GKM ring.
Proposition 7.12. There is a Q[a]-algebra and A × A-module isomorphism ← − R (x; a) → Ψ where the Q[a]-module structure is given by * . It sends Proof. Let f ∈ ← − R (x; a). Then f can be considered an element of Fun(S Z , Q[a]) by (4.5). For any The operators on ← − R (x; a) given by A x i , x i , A a i , and a i , correspond to the operators on Ψ given by A i •, a i •, A i * , and a i * respectively.
It is not hard to see that if f ∈ ← − R (x; a) is nonzero then it has a nonzero localization. Thus ← − R (x; a) embeds into Ψ. It suffices to show that ← − S v → ξ v for all v ∈ S Z . One may deduce that ← − S v (x; a) → ξ v by checking the conditions of Proposition 7.6. In turn, these can be verified by Proposition 4.7 and the analogue of Proposition 7.6 for S n , which is satisfied by the localizations of double Schubert polynomials S v (wa; a) for v, w ∈ S n . Restrcting Proposition 7.12 to S =0 -invariants gives the following result.
. It suffices to show that f | w (k) = 0 for some k. Let S be the finite set of indices i such that a i appears in some a λ . For sufficiently large k, the set I w,+ \ S has cardinality greater than deg(f ). If µ = (µ 1 , . . . , µ ℓ ) is minimal in dominance order in the set A = {λ | a λ = 0}, by Lemma 4.1 the polynomial p µ (x||a)| w (k) ∈ Q[a] contains a term of the form a µ a µ 1 r 1 a µ 2 r 2 · · · a µ ℓ r ℓ where r 1 > r 2 > · · · > r ℓ are the ℓ largest elements in I w,+ \S. This monomial does not appear in p λ (x||a)| w (k) for λ ∈ A\{µ}. The coefficient of this monomial must thus be nonzero in There is a partial multiplication map S 0 Z × S 0 Z → S 0 Z . The product of x ∈ S 0 Z and y ∈ S 0 Z is equal to z ∈ S 0 Z if (1) I x,+ ∩ I y,+ = ∅ = I x,− ∩ I y,− and (2) I x,± ∪ I y,± = I z,± . Proposition 7.15. There is a unique Hopf structure on Ψ Gr with comultiplication ∆ : Ψ Gr → Ψ Gr ⊗ Q[a] Ψ Gr given by (7.24) ∆(ψ)| x⊗y = ψ| xy whenever x, y, ∈ S 0 Z and xy ∈ S 0 Z is defined. With this Hopf structure, the map of Proposition 7.13 becomes a Q[a]-Hopf algebra isomorphism Λ(x||a) → Ψ Gr .
Proof. Suppose the product xy is well-defined. By Lemma 4.1, we have Thus (7.24) is consistent with the comultiplication of Λ(x||a). By the same argument as in the proof of Lemma 7.14, ∆(ψ) is completely determined by its values x ⊗ y for which xy is defined.

Peterson subalgebra
The Peterson subalgebra in the affine setting is recalled in Appendix C. LetS n be the affine symmetric group, generated by s i for i ∈ Z/nZ, with relations s 2 i = id, For i ∈ Z we write s i ∈S n for the element s i+nZ . We haveS n = Q ∨ ⋊ S n where Q ∨ is the coroot lattice, the Z-span of simple coroots α ∨ i = e i −e i+1 for 1 ≤ i ≤ n−1 where e i is a standard basis vector of Z n . For λ ∈ Q ∨ its image inS n is the translation element denoted t λ . LetS 0 n be the set of 0-Grassmannian elements those w ∈S n such that ws i > w for all i = 0. There is a bijectionS 0 n ↔ Q ∨ denoted w → t w . Suppose w ∈S 0 n and t w = t µ for µ ∈ Q ∨ . Let u µ ∈ S n be the shortest element such that u µ (µ) is antidominant. Then t w = t µ . = wu −1 µ .
8.1. Translation elements. The infinite symmetric group S Z does not contain translation elements. Nevertheless, it is possible to define elements τ w in the infinite nilHecke algebra which behave like translation elements. Recall that in Section 7.1 we have defined the nilHecke algebra A, which has a Q[a]-basis A w , w ∈ S Z . Let A ′ denote the completion of A, consisting of formal Q[a]-linear combinations of the elements A w . For a given w ∈ S Z , there are only finitely many pairs (u, v) ∈ S Z × S Z such that w . = uv. It follows that the multiplication in A induces a natural Q[a]-algebra structure on A ′ .
Recall also that we defined a comultiplication map ∆ : A. Under the pairing (7.23), A ′ is dual to Ψ. It follows from Proposition 7.12 that ∆ extends to a comultiplication such that for any sufficiently large n ≫ m, a reduced wordã for t w (treating w as an element of S n ) is obtained from a by the substitutions To explain how to find the above word, let Q ∨ Z ⊂ i∈Z Ze i be the infinite coroot lattice, the sublattice spanned by α (2.2)). There is a projection Q ∨ Z → Q ∨ denoted β → β, onto the translation lattice Q ∨ inS n , given by e i → e i+nZ . We have Let m be large enough so that λ is contained in the m × m square partition and n ≥ 2m. Then |I w λ ,± | ≤ m and all coordinates in β are in {−1, 0, 1} with coordinates 1 (resp. −1) occurring only in the first (resp. last) m positions. To prove Lemma 8.1 it suffices to show that the element u −1 β is in the image of a product of the generators (8.1) under the substitution (8.2). Since images of r and r ′ are inverses we may replace u −1 β by u β . It is enough to be able to sort β to antidominant using the generators. This is explained by the following example.
Each of these are infinite words in the alphabet {s i | i ∈ Z \{0}}, and each is a concatenation of two infinite reduced words. Abusing notation, we will use the same symbols r a,b and r b,a to represent the following permutations of Z (that do not belong to S Z ): Let S denote the set of infinite words in the alphabet {s i | i ∈ Z \ {0}} obtained as a finite concatenation of the words s i , i ∈ Z \ {0} and the words r a,b , a > 0 and b ≤ 0. Suppose a ∈ S, and s is a letter in a. Then we have a unique factorization a = a ′ s a ′′ where again a ′ , a ′′ ∈ S. We define a root β(s) by if the letter s is equal to s i . Here the action of a ′ on Q[a] is the one induced by the action on Z given by (8.5).
Definition 8.3. Let w ∈ S 0 Z . Define the infinite translation element τ w ∈ A ′ as follows. Take the word a of Lemma 8.1 and replace each occurrence of r or r ′ by infinite words as follows: r → r m,−m and r ′ → r −m,m to obtain an infinite word a w ∞ ∈ S. Now for v ∈ S Z , define ξ v | τ w ∈ Q[a] (cf. Proposition 7.3) by summed over finite subwords b of a w ∞ that are reduced words for v, and define τ w ∈ A ′ by Remark 8.4. Suppose w ∈ S 0 Z and we have I w,+ = {1 ≤ d t < d t−1 < · · · < d 1 } and I w,− = {e 1 < e 2 < · · · < e t ≤ 0}. Then a possible choice of a w ∞ is: where u is a reduced word for w and f j = |I w,− ∩ (−j, 0]| for j = 1, 2, . . . , t. Note that if w is the identity element, then τ w = 1.
Remark 8.5. In Definition 8.3 we have used Lemma 8.1 which relies on the notion of translation elements in the affine symmetric group. In future work we plan to study the Schubert calculus of a flag ind-variety associated to the affine infinite symmetric group Q ∨ Z ⋊ S Z , which contains translation elements τ w as defined above.
(1) For x ∈ S 0 Z , we have that τ x is a well-defined element of A ′ that does not depend on the choices of m and a in Lemma 8.1.
(4) We have τ x τ y = τ y τ x for any x, y ∈ S 0 Z . (5) We have τ x p = pτ x for any x ∈ S 0 Z and any p ∈ Q[a]. (6) We have ∆(τ x ) = τ x ⊗ τ x for any x ∈ S 0 Z . Proposition 8.6 is proven in Section 8.5.
8.2. The Peterson subalgebra. Let w Q(a)τ w denote the Q(a)-vector subspace of Q(a)⊗ Q[a] A ′ spanned by the elements τ w . Define the Q[a]-submodule P ⊂ A ′ by By Proposition 8.6(4), P lies within the centralizer subalgebra Z A ′ (Q[a]).
Recall that j w λ (a) denotes the coefficient of the double Schur function s λ (x||a) in the double Stanley symmetric function F w (x||a). For λ ∈ Y, define Theorem 8.7. For any λ ∈ Y, we have j λ ∈ P, and it is the unique element of P satisfying where a u ∈ Q[a] and the summation is allowed to be infinite. The submodule P is a free Q[a]-module with basis {j λ | λ ∈ Y}.
Theorem 8.7 will be proved in Section 8.6. Let P ′ be the completion of P whose elements are formal Q[a]-linear combinations of the elements {j λ | λ ∈ Y}. We call P ′ the Peterson subalgebra.
Theorem 8.8. The submodule P ′ ⊂ A ′ is a commutative and cocommutative Hopf algebra over Q[a].
Theorems 8.8 and 8.10 will be proved in Section 8.7.
Remark 8.11. Theorems 8.7, 8.8, and 8.10 hold over Z, but for consistency we work over Q.
8.3. Fomin-Stanley algebra. Let A denote the (infinite) nilCoxeter algebra, which is the Qalgebra with generators A i , i ∈ Z, satisfying the relations (2.6), (2.7), and (2.8). The algebra A has Q-basis A w , w ∈ S Z . Let A ′ denote the completion of A consisting of elements a = w a w A w that are infinite Q-linear combinations of the A w . Since every w ∈ S Z has finitely many factorizations of the form w . = xy, it follows that A ′ is a Q-algebra. There is a natural map φ 0 : A → A given by Define the Fomin-Stanley subalgebra B ⊂ A as the image φ 0 (P). Let j 0 λ := φ 0 (j λ ).
Theorem 8.12. The set {j 0 λ | λ ∈ Y} form a Q-basis of B. There is a Hopf-isomorphism B → Λ given by j 0 λ → s λ . Proof. Since {j λ | λ ∈ Y} form a basis of P, it is clear that {j 0 λ | λ ∈ Y} spans B. The equation (8.7) shows that j 0 λ = A w λ + other terms are linearly independent. The last statement follows from Theorem 8.8 and 8.10. Following [LaSh12] 2 , defineS 0 n ×S 0 n matricesÃ andB bỹ Both matricesÃ andB are lower-triangular when the rows and columns are ordered compatibly with the Bruhat order onS 0 n , and the entries belong to Q(a 1 , a 2 , . . . , a n ). For x ∈S n and v ∈S 0 n , denote byj x v ∈ Q[a 1 , a 2 , . . . , a n ] the affine double Edelman-Greene coefficient, and letj v ∈Ã denote the j-basis element (see Appendix C).
Proposition 8.13 ( [LaSc82]). Let v, w ∈S 0 n and x ∈S n . We have wv ξ x Fln (t w ). (8.10) Let ev n : Q[a] → Q[a 1 , a 2 , . . . , a n ] denote the Q-algebra morphism given by a i → a i mod n .
Proof. This follows from Proposition 7.3 which also holds in the affine case as well as the infinite case.
such that for all n ≫ 0, we haveÃ vw = ev n (A vw ) andB vw = ev n (B vw ).
Proof. Follows immediately from Lemma 8.14.
Lemma 8.16. Let x ∈ S Z and v ∈ S 0 Z . There exists a polynomial q(a) ∈ Q[a] such that for all n ≫ 0, we have thatj x v = ev n (q). Proof. Using Lemma 8.1 and Lemma 8.14, we deduce that for any u ∈ S Z and w ∈ S 0 Z , there is a polynomial p(a) ∈ Q[a] such that for sufficiently large n, we have ξ u Fln (w) = ev n (p(a)). By Proposition 8.13, we conclude that there is a polynomial q(a) ∈ Q[a] such thatj x v = ev n (q(a)) for sufficiently large n.
2 Our ξ v Fln |w differs from the one in [LaSc82] by a sign (−1) ℓ(v) 8.5. Proof of Proposition 8.6. Let x ∈ S 0 Z and v ∈ S Z . Only finitely many subwords of a x ∞ are reduced words for v, and for n ≫ 0 there is a bijection between such subwords and subwords ofã that are reduced words for v (now thought of as an element inS n ). It thus follows from the definitions that for n ≫ 0 we have (8.11) ev n (ξ v (τ x )) = ξ v Fln (t x ). Claim (1) follows immediately. Claim (2) follows from the similar claim in the affine nilHecke algebraÃ.
Let x, y ∈ S 0 Z and v ∈ S Z . Only finitely many pairs of terms from the expansion (8.6) for τ x and τ y contribute to the coefficient of A v in the product τ x τ y . Thus for n ≫ 0 the coefficient of A v in τ x τ y is taken to the coefficient of A v in t x t y by ev n . Claims (3) and (4) now follow from similar statements in the affine case (see (C.1)).
Let x ∈ S 0 Z , v ∈ S Z , and p ∈ Q[a]. Only finitely many terms of the expansion (8.6) for τ x contribute to the coefficient of A v in τ x p. Thus for n ≫ 0 the coefficient of A v in τ x p is taken to the coefficient of A v in t x ev n (p) by ev n . Claim (5) now follows from (C.2) in the affine case.
Let v ∈ S Z . Then for n ≫ 0, the calculation of ∆(A v ) in the affine nilHecke ringÃ is identical to that in A. Claim (6) now follows from the equality ∆(t x ) = t x ⊗ t x in the affine case (see (C.3)).
Proof. By Theorem 10.9, the image of By Proposition 8.17, the element j λ ∈ A ′ is the limit (taking limits of coefficients of A v ) of j w λ ∈Ã as n → ∞. By (8.11), the element τ w ∈ A ′ is a similar limit of the elements t w ∈Ã. Combining Lemma 8.15 and (8.9), we thus conclude that It follows that j λ ∈ P. The expansion (8.7) follows from Theorem C.1.
By Lemma 8.15 and (8.8), we have that both {j λ | λ ∈ Y} and {τ w | w ∈ S 0 Z } form bases of Q(a) ⊗ Q[a] P. Thus an arbitrary element of a = aw A w ∈ P is uniquely determined by the coefficients {a w ∈ Q[a] | w ∈ S 0 Z }. Indeed, we have a = λ∈Y a w λ j λ and the sum must be finite. It follows that P is a free Q[a]-module with basis {j λ | λ ∈ Y}. 8.7. Proof of Theorems 8.8 and 8.10.
Proposition 8.18. For λ, ν ∈ Y, we have Proof. Let us calculate the coefficient of s λ (x||a) ⊗ s µ (x||a) in ∆(F w (x||a)). On the one hand, by Corollary 4.16. So the coefficient is equal to ν⊃µ j w ν/µ λ j w ν , which is the coefficient of A w on the RHS of (8.12).
On the other hand, by Corollary 4.15, we have So the coefficient is also equal to w . =uv j u λ j v µ , which (using Proposition 8.6(5) to obtain that j λ commutes with Q[a]) is equal to the coefficient of A w on the LHS of (8.12).
It follows from Proposition 8.18 that P ′ is a commutative Q[a]-algebra. Together with Proposition 8.6(6), we obtain Theorem 8.8.
The pairing (7.23) induces a pairing between P ′ and Ψ Gr . By Proposition 7.11(3), we have ξ v , j λ = δ vw λ for v ∈ S 0 Z and λ ∈ Y. Thus P ′ and Ψ Gr are dual Q[a]-modules. By Proposition 7.11(2), the comultiplication in P ′ is dual to the multiplication in Ψ Gr . By comparing Proposition 8.18 and Corollary 4.16, the multiplication of P ′ is dual to the multiplication in Ψ Gr . Thus P ′ and Ψ Gr are dual Q[a]-Hopf algebras. By Proposition 7.15 and the definition of s λ (y||a), we have an induced isomorphism of Q[a]-Hopf algebras P ′ ∼ =Λ(y||a) sending j λ to s λ (y||a). This completes the proof of Theorem 8.10.
The resulting polynomial must equal j x v (a). By Theorem C.2, we have thatj x v (a) is a positive integer polynomial expression in the linear forms a 1 − a 2 , a 2 − a 3 , . . . , a n−1 − a n .
Applying the above substitution to this expression gives the desired expression for j x v (a).

Back stable triple Schubert polynomials
In this section we define triple back stable Schubert polynomials and triple Stanley symmetric functions. This allows effective computation of some double Edelman-Greene coefficients and structure constants for dual Schur functions. 9.1. Tripling. Let ν a,b : Λ(a) → Λ(b) be the map that changes symmetric functions from the a-variables to b-variables. Let Λ(x/b) ⊂ Λ(x) ⊗ Q Λ(b) denote the image of the superization map p k → p k (x/b). We use the same notation for the Q[a]-algebra maps These maps change a's to b's but only "in symmetric functions". All of these maps are Q[a]algebra isomorphisms: the inverse is the substitution f → f | b=a . Finally, note that we have an injection Λ(x||a) ֒→ Λ(x) ⊗ Λ(a) ⊗ Q[a], and the action of ν a,b on Λ(x||a) is simply given by 1 ⊗ ν a,b ⊗ 1 : 9.2. Back stable triple Schubert polynomials. For w ∈ S Z , define the back stable triple Schubert polynomials [a]. In particular, the structure constants for ← − S w (x; a; b) (which are equal to the structure constants for ← − S w (x; a)) belong in Q[a].
Proof. The first equality follows from applying ν a,b to (4.9). The second equality follows from applying ν a,b to Proposition 4.3.
Recall that A x i (resp. A a i , A b i ) denotes the divided difference operator in the x-variables (resp. a-variables, b-variables).
Proposition 9.2. For w ∈ S Z and i ∈ Z, we have Proof. The first statement follows immediately from the last equality in Proposition 9.1 and Theorem 3.2. For i = 0, we have A a i • ν a,b = ν a,b • A a i , so the second statement follows by Proposition 4.8. 9.3. Triple Stanley symmetric functions. Define the triple Stanley symmetric functions by F w (x||a||b) := ν a,b (F w (x||a)).
Recall the Q-algebra automorphism γ a of §4.5. This map can be applied to the Q[a]-algebra x, a] or to the Q[a]-algebra Q [a, b]. Recall also γ : S Z → S Z from §2.1.
Proof. Follows from Lemma 9.3 and Corollaries 4.4 and 3.8.
Corollary 9.5. Let λ ∈ Y and w ∈ S Z . Then j γ(w) λ (a, b) = γ a (j w λ (a, b)). Thus triple Stanley symmetric functions allow us to distinguish between "stable" phenomena (the b-variables) and the "shifted" phenomena (the a-variables).
9.4. Double to triple. We have an explicit formula for j w λ (a, b) in terms of double Edelman-Greene coefficients j w λ (a). Recall the definition of Durfee square d(λ) from before Proposition 4.23. Proposition 9.6. Let λ, µ ∈ Y. Then For µ ∈ Y and w ∈ S Z we have Proof. By Proposition 4.23, we have This gives the first formula. The second formula follows from (9.2).
The following result follows from (9.3).
Proposition 9.7. Let w ∈ Y, w ∈ S Z , and i ∈ Z − {0}. Then if µ has no addable box on diagonal i. 9.5. Triple Stanley coefficients for a hook. In this section we compute j w (q+1,1 p ) (a, b) for all w ∈ S Z and p, q ≥ 0, in a way that exhibits the positivity of Theorem 4.20.
The support of a permutation w ∈ S Z is the finite set of integers |w| := {i | s i appears in a reduced word of w} ⊂ Z.
A permutation w ∈ S Z is called increasing (resp. decreasing) if it has a reduced word s i 1 s i 2 · · · s i ℓ such that i 1 < i 2 < · · · < i ℓ (resp. i 1 > i 2 > · · · > i ℓ ). For J ⊂ Z a finite set, we denote by u J ∈ S Z (resp. d J ∈ S Z ) the unique increasing (resp. decreasing) permutation with support J. We call w ∈ S Z a Λ if it has a factorization of the form w . = u J d K . Such factorizations are called Λ-factorizations. We consider two factorizations to be distinct if their pairs (J, K) are distinct. We call a reduced word u a Λ-word if it is first increasing then decreasing. Associated to a Λfactorization is a unique Λ-reduced word.
Suppose w admits a nontrivial Λ-factorization id = w .
There are exactly two pairs (J, K) corresponding to a given pair (J ′ , K ′ ): m occurs in exactly one of J and K.
Theorem 9.8. Let p, q ≥ 0 and w ∈ S Z . Then j w λ (a, b) = 0 unless w is a Λ, in which case where the sum runs over all distinct pairs (J ′ , K ′ ) coming from Λ-factorizations w = u J d K and Remark 9.9. The coefficients j w λ (a, b) appear to satisfy the following generalization of the positivity in Theorem 4.20: j w λ (a, b) is a sum of products of binomials c − d where c and d are variables with The double Schubert polynomials occurring in Theorem 9.8 satisfy this positivity, say, by the formula for the monomial expansion of double Schubert polynomials in [FK].
Remark 9.10. It is possible to obtain more efficient formulas than those in Theorem 9.8, especially when p = q = 0, by grouping terms according to the set of maxima for each of the maximal subintervals of |w|. More generally, for w = s i s i+1 · · · s k , we have j w Let θ = a 1 − a 0 . The proof of Theorem 9.8 uses localization formulas for Schubert classes in equivariant cohomology H * Tn ( Fl n ) (see §10.7) of the affine flag variety. In this context we set a i = a i+n for all i ∈ Z. We shall use the following result [LaSh12,Theorem 6].
Theorem 9.12. For every id = x ∈ S n , we have θ −1 ξ x −1 Fln | s θ ∈ Q[a 1 , a 2 , . . . , a n ] and Lemma 9.13. Let id = x ∈ S n . Then ξ x Fln | s θ = 0 unless x is a Λ, in which case Proof. We compute ξ x Fln | s θ as an element of Q[a 0 , a 1 , . . . , a n−1 ] (setting a n = a 0 ), using Proposition 7.3, picking the reduced word u = s 1 s 2 · · · s n−1 · · · s 2 s 1 of s θ . If x has no Λ-factorization, then u does not contain a reduced word for x. For i = n − 1, the roots β(s i ) associated to s i are a i+1 − a 1 (left occurrence) and a 0 − a i+1 (right occurrence), the sum of which is a 0 − a 1 . We also have β(s n−1 ) = a 0 − a 1 .
Summing over the Λ-factorizations, the simple generator s m where m = max(|w|) contributes a factor of (a 1 − a 0 ) to (−1) ℓ(x) ξ x | s θ . The remaining simple generators contribute j∈J ′ (a 1 − a j+1 ) k∈K ′ (a k+1 − a 0 ). Finally, these products of binomials are double Schubert polynomials: Proof of Theorem 9.8. First suppose that w ∈ S + . Combining Theorem 9.12 and Lemma 9.13 with the limiting arguments of Section 8.4, we deduce (noting that Theorem 9.12 has "x −1 ") that Recall the shift automorphism γ : S Z → S Z from §2.1. It follows that we must have to be consistent with Corollary 9.5, and this must hold for all w ∈ S Z . The formula for a general hook (q + 1, 1 p ) follows by Proposition 9.7. 9.6. Proof of Theorem 6.14. By Theorem 8.10 and Proposition 8.18, the coefficient ofŝ λ (y||a) in the productŝ 1 (y||a)ŝ µ (y||a) is equal to 0 if µ ⊆ λ and equal to j w λ/µ 1 (a) otherwise.

Infinite flag variety
10.1. Infinite Grassmannian. We define the Sato Grassmannian Gr · . Let F = C((t)) be the space of Laurent power series with coefficients in C. For an integer a ∈ Z, The Sato Grassmannian Gr · is the set of all admissible subspaces in F and we have Gr · = k Gr (k) , where Gr (k) consists of admissible subspace with virtual dimension k. We will mostly focus our attention on the virtual dimension 0 component, Gr := Gr (0) , which we call the infinite Grassmannian.
The infinite Grassmannian Gr has the structure if an ind-variety over C. There is a bijection between {Λ | E N ⊂ Λ ⊂ E −N and vdim(Λ) = 0} and the points of the finite-dimensional Gr(N, 2N ). We have Gr = N Gr(N, 2N ), from which Gr inherits the structure of an ind-variety over C.
10.2. Infinite flag variety. An admissible flag (of virtual dimension 0) in F is a sequence of admissible subspace satisfying the conditions: (1) vdim(Λ i ) = i, and (2) for some N , we have The infinite flag variety Fl is the set of all admissible flags. We have Fl = N Fl(2N ), from which Fl inherits the structure of an ind-variety over C. We have projection maps π i : Fl → Gr (i) given by Λ • → Λ i .
We also have the shifted infinite flag varieties Fl (k) where we impose the condition that vdim(Λ i ) = i + k. Each Fl (k) is isomorphic to Fl, and we have the Sato flag variety Fl · = k Fl (k) .
10.3. Schubert varieties. Let B be the group of C-linear transformations of F generated by T Z and by elements x i,j (a) for i < j and that act on F by The group B acts on Fl. For each w ∈ S Z , the Schubert cell Ω w := B · w is isomorphic to the affine space C ℓ(w) and is completely contained in Fl(2N ) if w ∈ S [−N,N −1] . We define the Schubert variety X w to be the closure It has dimension ℓ(w), and we have X w = v≤w Ω v .
10.4. Equivariant cohomology of infinite flag variety. All our cohomologies have Q-coefficients. We consider the torus T Z = (C × ) Z which is defined to be the restricted product, where all but finitely many factors must be the identity. Then T Z is homotopy equivalent to the restricted product (S 1 ) Z , which is easily seen to be a CW-complex of infinite dimension and with infinitely many cells in each dimension. We then have that ET Z is (S ∞ ) Z , which is again a restricted product where all but finitely many factors must be the basepoint of S ∞ . The classifying space The image Λ • = ζ(L • ) is a flag of admissible subspaces in F . However, it is not an admissible flag since it is possible that ζ(L i ) = E i for infinitely many i ∈ Z. We do not have an embedding of Fl · n in the Sato flag variety Fl · . Nevertheless, Fl · n is known to be an ind-variety over C [Kum]. 10.7. Equivariant cohomology of affine flag variety. Let T n be the maximal torus of GL n (C). We have H * Tn (pt) ∼ = Q[a 1 , . . . , a n ]. Write γ a : Q[a] → Q[a] for the Q-algebra isomorphism given by a i → a i+1 mod n .
The torus T n acts on Gr n and Fl n . LetS n be the affine Coxeter group of SL n (C) and S n = Z ⋉S n = sh ×S n the affine Weyl group of GL n (C). For w ∈ S n , let ξ w There is a wrong way map [Lam, LSS] ̟ : H * Tn ( Fl n ) → H * Tn ( Gr n ) induced by the homotopy equivalences ΩSU (n) ∼ = Gr n and LSU (n)/T n ∼ = Fl n , and the inclusion ΩSU (n) ֒→ LSU (n)/T n . The class ̟(ξ) is completely determined by its localization at T n -fixed points of Gr n : 10.8. Presentations. We have a ring map ev n : H * T Z (pt) → H * Tn (pt) which sets equal a i = a i+n for all i ∈ Z.
The inclusion Gr n ֒→ Gr induces a map of H * Tn (pt)-algebras: To explain this, we would like to embed T n into T Z in an n-periodic manner, but our definition of T Z requires all but finitely many entries to be identity. However, the action of T n on Gr n is compatible with the action of T Z on Gr as follows. Take N = mn for some positive integer m. If we restrict ourselves to the finite-dimensional piece k Gr(k, 2N ) of Gr, then the action of T Z factors through T [−N,N −1] , and this is the same as the action of T n on Gr n ∩ ( k Gr(k, 2N )) where we embed T n into T [−N,N −1] in a n-periodic manner. Thus the embedding Gr n → Gr is "essentially" T n -equivariant, and induces (10.2) by pullback.
Proof. LetΨ n ⊂ Fun(S n , Q[a 1 , . . . , a n ]) denote the image of H * Tn ( Fl n ) under localization. It is given by GKM conditions similar to (7.9). It is straightforward to check that the generators x i and p k (x||a) of ← − R (x; a) evn are sent toΨ n under the diagonal map f (x; a) → (w → f (wa; a)). Furthermore, this diagonal map is clearly a Q[a 1 , . . . , a n ] algebra morphism. This uniquely determines the map φ n with the desired properties.
In fact, the map φ n is a surjection and gives a presentation of the cohomologies H * Tn ( Fl n ) and H * Tn ( Gr n ). We shall study these presentations in further detail in [LLSb].
Remark 10.7. The map φ n cannot be induced by any continuous map Fl n → Fl that sends T nfixed points to T Z -fixed points. This is because for any w ∈S n and v ∈ S Z , one can always find f (x; a) ∈ ← − R (x; a) such that f (wa; a) = f (va; a).
Proposition 10.8. We have a commutative diagram Proof. By (10.1) and Proposition 10.6, it suffices to check that for f (x; a) ∈ ← − R (x; a) evn and λ ∈ Q ∨ , we have For f ∈ Λ(x||a), a formula for f (t λ a; a) is given in [LaSh13,Section 4.5]. For p ∈ Q[x, a], we have t λ x i = x i + λ i δ = x i (since we are working with the finite, or level zero, torus T n rather than the affine one). Thus p(t λ a; a) = η a (p) for p ∈ Q[x, a] and (10.5) holds.
10.9. Small affine Schubert classes. We shall need the following result from [LLSb] concerning "small" affine Schubert polynomials.
Theorem 10.9. Suppose that w ∈ S Z (resp. w ∈ S 0 Z ), which we also consider an element ofS n (resp.S 0 n ) for n ≫ 0. For sufficiently large n ≫ 0 the image of Proof. We sketch the proof. There are divided difference operators Aī : H * Tn ( Fl n ) → H * Tn ( Fl n ) for i ∈ Z/nZ, and the Schubert classes ξ w Fln are determined by recurrences similar to (7.19). One then checks that for Schubert classes indexed by small w ∈ S Z , the action of A i on ← − R (x; a) and on H * Tn ( Fl n ) are compatible: Aī • φ n = φ n • A i acting on ← − S w (x; a), when i ∈ Z is chosen carefully. It follows that ← − S w (x; a) represents ξ w Fln for sufficiently large n.

Graph Schubert varieties
11.1. Schubert varieties and double Schur functions. Fix a positive integer n. Let Gr(n, 2n) denote the Grassmannian of n-planes in C 2n = span(e 1−n , e 2−n , . . . , e n ). We let the torus T 2n = (C × ) 2n act on C 2n , and identify H * T 2n (pt) = Q[a 1−n , a 2−n , . . . , a n ], so that the weight of the basis vector e i ∈ C 2n is equal to a i . The T -fixed points of Gr(n, 2n) are the points e I ∈ Gr(n, 2n), where I is an n-element subset I ⊂ [1 − n, n]. There is a bijection from partitions λ fitting in a n × n box to [1−n,n] n given by λ → I(λ) = ([1, n] \ S + ) ∪ S − , where λ = λ(S − , S + ); see §2.1. The Schubert variety X λ has codimension |λ| and contains the T -fixed points e I(µ) for µ ⊇ λ. Via the forgetful map Q[a] → H * T 2n (pt) which sets a i to 0 for i / ∈ [1 − n, n], H * T 2n (pt) ⊗ Q[a] Λ(x||a) has a Q[a 1−n , . . . , a n ]-algebra structure. 3 1 2 3 4 (1) (2) (3)(4)(5) Going through the definition of IP pipedream in [Knu14], we see that they are in bijection with rectangular w-bumpless pipedreams. Comparing wt(D) with wt(P ), it follows from Proposition 11.1 and Theorem 11.3 that in H * T 2n (Gr(n, 2n)) we have (11.3) F (n) w (x||a) = P wt (n) (P )s λ(P ) (x||a), where the summation is over all rectangular w-bumpless pipedreams, and wt (n) (P ) = wt(P )| x i →a i−n . But we have injections S n ֒→ S n+1 ֒→ · · · . The rectangular S n+1 -bumpless pipedreams P ′ for w are obtained from the rectangular S n -bumpless pipedreams P for w by (1) adding an elbow in the southeastern most corner, (2) filling the rest of the southmost row with vertical pipes, and (3) filling the rest of the eastmost column with horizontal pipes. Thus, (11.3) holds for all sufficiently large n, where the summation is over the same set of rectangular w-bumpless pipedreams. The only expansion of ← − S(x; a) in terms of s λ (x||a) consistent with this is the one in Theorem 5.11. 11.4. Proof of Theorem 11.3. There is an embedding ι : Gr(n, 2n) → Gr (n) 2n , placing the Grassmannian as a Schubert variety at the "bottom" of the affine Grassmannian of GL 2n . This induces a pullback back map ι * : H * T 2n ( Gr (n) 2n ) → H * T 2n (Gr(n, 2n)). There is also the wrongway map of rings ̟ : H * T 2n ( Fl ). For a bounded affine permutation f , let [Π f ] ∈ H * T 2n (Gr(n, 2n)) denote its equivariant cohomology class, and let ξ f ∈ H * T 2n ( Fl (n) 2n ) denote the Schubert class. The following result is due to Knutson-Lam-Speyer [KLS] (see also He and Lam [HL]).
In particular, this result holds for Π f = Πf w = G(w). The remainder of the proof is concerned with working through the interpretation of Theorem 11.4 in terms of double symmetric functions.
Theorem 11.3 follows from this equality and Theorem 11.4. 11.5. Divided difference formula for graph Schubert class. For completeness, we include the following formula due to Allen Knutson.
Theorem 11.5. Let w ∈ S n . Then where the action of A w 0 is defined by the action of S n on the variables x 1−n , . . . , x −1 , x 0 .
By [BF] this map has a section σ : H * T 2n (Gr(n, 2n)) → H * GLn×T 2n (M n×2n ) such that for any closed subscheme Z ⊂ Gr(n, 2n), σ([Z]) = [π −1 (Z)]. In particular σ([X λ ]) = [π −1 (X λ )] which is identified with the double Schur polynomial S [1−n,n] w λ (x; a) in variables x 1−n , . . . , x 0 and a 1−n , . . . , a n where the row torus T n ⊂ GL n acts on M n×2n by the weights x 1−n through x 0 and T 2n acts on columns by weights a 1−n through a n . Let Z = G(w) and Y = π −1 (G(w)) ⊂ M n×2n . In the notation of §11.2 we have σ ([G(w) Since B + acts freely on (B − |M • w ) one may show that [Y ] = A w 0 [Y ′ ] where [Y ′ ] ∈ H * Tn×T 2n (M n×2n ). But Y ′ is a product. The equivariant class of the affine space B − is the product of the weights of the matrix entries that are set to zero in B − and the equivariant class of M w is γ −n x S w (x; a) by [KM] (the shift in x variables is due to the convention on weights). We deduce that as required.
Proof. We expand using the Billey-Jockusch-Stanley formula (2.12): w . =uy (−1) ℓ(y) S u −1 (x)S y (x) = a 1 a 2 ···a ℓ ∈R(w) We perform a sign-reversing involution on the inner sum on the RHS (contained inside the parentheses) as follows. If either (k > 0 and b k < b k+1 ) or k = ℓ, then we change k to k − 1. If either (k < ℓ and b k > b k+1 ) or k = 0, then we change k to k + 1. If 0 < k < ℓ and b k = b k+1 , then we change k to k − 1 if a k < a k+1 ; we change k to k + 1 if a k > a k+1 .
B.2. Inverting systems with Schubert polynomials as change-of-basis matrix. Let W ⊂ S Z be a subgroup generated by simple reflections r i for i ∈ I for some I ⊂ Z. For J ⊂ I let W J be the subgroup of W generated by r i for i ∈ J. For x, y ∈ W say x J ≤ y if yx −1 ∈ W J and ℓ(yx −1 ) + ℓ(x) = ℓ(y). Equivalently, x J ≤ y if and only if there is a v ∈ W J such that y . = vx.
are mutually inverse.
The right hand analogue also holds. For x, y ∈ W say x ≤ J y if x −1 y ∈ W J and ℓ(x) + ℓ(x −1 y) = ℓ(y). Equivalently, x ≤ J y if and only if there is a v ∈ W J such that y . = xv.
Lemma B.2. Let W ′ be a fixed coset of W/W J . The W ′ × W ′ -matrices are inverses.
Corollary B.3. Let {F w | w ∈ W } and G w | w ∈ W } be families of elements. Then for all w ∈ W . (B.4) Appendix C. Level zero affine nilHecke ring We recall in this section standard results concerning the affine symmetric group, the affine nilHecke algebra, and the Peterson subalgebra. C.1. Affine symmetric group. The affine symmetric groupS n is the infinite Coxeter group with generators s 0 , s 1 , . . . , s n−1 and relations s i s j = s j s i for |i − j| ≥ 2 and s i s i+1 s i = s i+1 s i s i+1 for all i. Here indices are taken modulo n.
We have an isomorphismS n ∼ = S n ⋊ Q ∨ , where Q ∨ := {λ = (λ 1 , . . . , λ n ) | n i=1 λ i = 0} ⊂ Z n . For λ ∈ Q ∨ , we write t λ ∈S n for the corresponding translation element. Then (C.1) t λ t µ = t λ+µ = t µ t λ and wt λ w −1 = t w·λ . Each coset wS n for S n insideS n contains a unique translation element t w , and a unique affine Grassmannian element i.e. a coset representative x ∈ wS n that is minimal length in wS n . We denote byS 0 n ⊂S n the affine Grassmannian elements. C.2. Level zero affine nilHecke ring. LetÃ denote the level zero affine nilHecke ring (see for example [LaSh12] for details). It has Q[a 1 , a 2 , . . . , a n ]-basis {A w | w ∈S n }. There is an injectioñ S n ֒→Ã that is a group isomorphism onto its image. It is given by s i → 1−α i A i = 1−(a i+1 −a i )A s i . The image ofS n inÃ forms a basis ofÃ over Q(a 1 , a 2 , . . . , a n ).
Theorem C.1. The Peterson subalgebraP is a commutative subalgeba ofÃ. It is a free Q[a 1 , . . . , a n ]module with basis {j λ | λ ∈ Q ∨ }. The elementj λ ∈P is uniquely characterized by the expansioñ forj u λ ∈ Q[a 1 , . . . , a n ], where wS n = t λ S n . The following result follows from combining [LaSh10], which proves Peterson's isomorphism of localizations of H * ( Gr) and the equivariant quantum cohomology H * Tn (Fl n ) together with an explicit correspondence of Schubert classes, and the positivity result of [Mih] in equivariant quantum cohomology.