Quantum Double Schubert Polynomials Represent Schubert Classes

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 3, March 2014, Pages 835–850 S 0002-9939(2013)11831-9 Article electronically published on December 11, 2013

QUANTUM DOUBLE SCHUBERT POLYNOMIALS REPRESENT SCHUBERT CLASSES

THOMAS LAM AND MARK SHIMOZONO

(Communicated by Lev Borisov)

Abstract. The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schu- bert classes in Kim’s presentation of the equivariant quantum cohomology of the flag variety. Parabolic analogues of quantum double Schubert polynomials are introduced and shown to represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. This establishes a new method for computing equivariant Gromov-Witten invariants for partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.

1. Introduction Let H∗(Fl), HT (Fl), QH∗(Fl), and QHT (Fl) be the ordinary, T -equivariant, quantum, and T -equivariant quantum cohomology rings of the variety Fl = Fln of n complete flags in C where T is the maximal torus of GLn. All cohomologies are with Z coefficients. The flag variety Fln has a stratification by Schubert varieties Xw, labeled by permutations w ∈ Sn, which gives rise to Schubert bases for each of these rings. The structure constants of these bases play an important role in classical and modern enumerative geometry. For QHT (Fl) they are known as the 3-point genus 0 Gromov-Witten invariants. This paper is concerned with the problem of finding polynomial representatives for the Schubert bases in a presentation of these (quantum) cohomology rings. These ring presentations are due to Borel [Bo] in the classical case, and to Ciocan- Fontanine [Cio], Givental and Kim [GK] and Kim [Kim] in the quantum case. This problem has been solved in the first three cases: the Schubert polynomials ∗ are known to represent Schubert classes in H (Fln) by work of Bernstein, Gelfand, and Gelfand [BGG] and Lascoux and Schützenberger [LS]; the double Schubert polynomials, also due to Lascoux and Schützenberger, represent Schubert classes T in H (Fln) (see for example [Bi]); and the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov [FGP] represent Schubert classes in QH∗(Fl). These polynomials are the subject of much research by combinatorialists and geometers, and we refer the reader to these references for a complete discussion of these ideas. Our first main result (Theorem 3.4) is that the quantum double Schubert polynomials of [KM, CF] represent equivariant quantum Schubert classes in QHT (Fl). Anderson and Chen [AC] have announced a proof using the geometry of Quot schemes.

Received by the editors October 22, 2011 and, in revised form, April 17, 2012. 2010 Mathematics Subject Classiﬁcation. Primary 14N35; Secondary 14M15. The ﬁrst author was supported by NSF grant DMS-0901111 and by a Sloan Fellowship. The second author was supported by NSF DMS-0652641 and DMS-0652648.

c 2013 American Mathematical Society Reverts to public domain 28 years from publication 835

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Let SLn/P be a partial flag variety, where P denotes a parabolic subgroup of SLn. In nonquantum Schubert calculus, the functorality of (equivariant) cohomology implies that the (double) Schubert polynomials labeled by minimal length coset ∗ T representatives again represent Schubert classes in H (SLn/P )orH (SLn/P ). This is not the case in quantum cohomology. Ciocan-Fontanine [Cio2] solved ∗ the corresponding problem in QH (SLn/P ), extending Fomin, Gelfand, and Post- nikov’s work to the parabolic case. Here we introduce parabolic quantum double Schubert polynomials. Our second main result asserts that these polynomials repre- T sent Schubert classes in the torus-equivariant quantum cohomology QH (SLn/P ) of a partial flag variety. Earlier, Mihalcea [Mi2] had observed that the double (or factorial) Schur polynomials represent the Schubert basis in the special case of the equivariant quantum cohomology of the Grassmannian. Our results imply that the equivariant 3-point genus 0 Gromov-Witten invariants for partial flag varieties coincide with the structure constants for the parabolic quantum double Schubert polynomials. Moreover no additional computation is qa § required for reduction modulo the ideal (denoted JP in 6.2): a parabolic quantum double Schubert polynomial either projects to a Schubert basis element or to zero, according as the indexing permutation is in the correct symmetric group or not. Our proof relies on Mihalcea’s characterization [Mi] of the equivariant quantum Schubert classes and ring by his equivariant quantum Chevalley-Monk formula (Theorems 2.1 and 6.11).

2. The (quantum) cohomology rings of flag manifolds

2.1. Presentations. Let x =(x1,...,xn), a =(a1,...,an), and q =(q1,...,qn−1) be indeterminates. We work in the graded polynomial ring Z[x; q; a] with deg(xi)= T deg(ai) = 1 and deg(qi)=2.LetS = Z[a]beidentiﬁedwithH (pt) and let ej (a) be the elementary symmetric polynomial. Let Cn be the tridiagonal n × n matrix with entries xi on the diagonal, −1 on the superdiagonal, and qi on the subdiagonal. n ∈ Z Deﬁne the polynomials Ej [x; q]by n − − n−j n det(Cn t Id) = ( t) Ej . j=0 Let J (resp. J a, J q, J qa) be the ideal in Z[x](resp. S[x], Z[x; q], S[x; q]) generated − n n − ≤ ≤ by the elements ej (x)(resp. ej (x) ej (a); Ej ; Ej ej (a)) for 1 j n;inall cases the j-th generator is homogeneous of degree j.Wehave ∗ ∼ (2.1) H (Fl) = Z[x]/J, T ∼ a (2.2) H (Fl) = S[x]/J , ∗ ∼ q (2.3) QH (Fl) = Z[x; q]/J , T ∼ qa (2.4) QH (Fl) = S[x; q]/J as algebras over Z, S, Z[q], and S[q] respectively. The presentation of H∗(Fl) is a classical result due to Borel. The presentations of QH∗(Fl) and QHT (Fl) are due to Ciocan-Fontanine [Cio], Givental and Kim [GK] and Kim [Kim].

2.2. Schubert bases. Let Xw = B−wB/B ⊂ Fl be the opposite Schubert variety, where w ∈ W = Sn is a permutation, B ⊂ SLn is the upper triangular Borel and

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∗ T B− the opposite Borel. The ring H (Fl) (resp. H (Fl)) has a basis over Z (resp. S), denoted [Xw](resp.[Xw]T ), associated with the Schubert varieties. Given three elements u, v, w ∈ W and an element of the coroot lattice β ∈ Q∨ w ∈ Z one may define a genus zero Gromov-Witten invariant cuv(β) ≥0 (see [GK,Kim]) and an associative ring QH∗(Fl) with Z[q]-basis {σw | w ∈ W } (called the quantum Schubert basis) such that u v w w σ σ = qβcuv(β)σ , w,β − − n 1 ki n 1 ∨ ∈ Z where qβ = i=1 qi where β = i=1 kiαi for ki . Similarly there is a basis of a ring QHT (Fl) with S[q]-basis given by the equivariant quantum Schubert classes w σT , defined using equivariant Gromov-Witten invariants, which are elements of S. We shall use the following characterization of QHT (Fl) and its Schubert basis { w | ∈ } + σT w W due to Mihalcea [Mi]. Let Φ be the set of positive roots and ρ the half sum of positive roots. For w ∈ W define

+ Aw = {α ∈ Φ | wsα w}, + ∨ Bw = {α ∈ Φ | (wsα)=(w)+1−α , 2ρ}. a ··· ∈ n n Let ωi = a1 + + ai S be the fundamental weight. We write Aw and Bw to emphasize that the computation pertains to Fl = SLn/B.

Theorem 2.1 ([Mi, Corollary 8.2]). For w ∈ Sn and 1 ≤ i ≤ n − 1 aDynkin w node, the equivariant quantum Schubert classes σT satisfy the equivariant quantum Chevalley-Monk formula s ∨ ws ∨ ws i w − a · a w α ∨ α (2.5) σT σT =( ωi + w ωi ) σT + α ,ωi σT + qα α ,ωi σT . ∈ n ∈ n α Aw α Bw { w | ∈ } Moreover these structure constants determine the Schubert basis σT w Sn and T the ring QH (SLn/B) up to isomorphism as S[q]-algebras.

3. Quantum double Schubert polynomials

Now we work with inﬁnite sets of variables x =(x1,x2,...), q =(q1,q2,...), and a =(a1,a2,...).

a −1 − a 3.1. Various Schubert polynomials. Let ∂i = αi (1 si ) be the divided dif- − a ference operator, where αi = ai ai+1 and si is the operator that exchanges ai a and ai+1. Since the operators ∂i = ∂i satisfy the braid relations one may deﬁne ··· ··· ∈ ∂w = ∂i1 ∂i ,wherew = si1 si is a reduced decomposition. For w Sn deﬁne the double Schubert polynomial Sw(x; a) [LS] and the quantum double Schubert polynomial S˜ w(x; a) ∈ S[x; q][KM,CF]by n−1 i (n) − (ww0 ) a − (3.1) Sw(x; a)=( 1) ∂ (n) (xj an−i), ww0 i=1 j=1 n−1 (n) ˜ − (ww0 ) a − (3.2) Sw(x; a)=( 1) ∂ (n) det(Ci an−iId), ww0 i=1

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(n) ∈ 1 where w0 Sn is the longest element. Note that it is equivalent to define ˜ Sw(x; a) by setting the qi variables to zero in Sw(x; a). Let S∞ = n≥1 Sn be the infinite symmetric group under the embeddings in : Sn → Sn+1 that add a fixed point at the end of a permutation. Due to the stability ˜ ˜ ∈ property [KM] Sin(w)(x; a)=Sw(x; a)forw Sn, the quantum double Schubert ˜ polynomials Sw(x; a) are well-defined for w ∈ S∞. Similarly, Sw(x; a) is well- defined for w ∈ S∞. For w ∈ S∞, define the (resp. quantum) Schubert polynomial Sw(x)=Sw(x;0) (resp. S˜ w(x)=S˜ w(x; 0)) by setting the ai variables to zero in the (resp. quantum) ˜ ˜ double Schubert polynomial. Note that Sw(x), Sw(x; a), Sw(x), and Sw(x; a)are all homogeneous of degree (w). The original definition of a quantum Schubert polynomial in [FGP] is different. However their definition and the one used here are easily seen to be equivalent [KM] due to the commutation of the divided differences in the a variables and the quantization map θ of [FGP], which we review in §3.3.

Lemma 3.1 ([Mac]). For w ∈ S∞, the term that is of highest degree in the x variables and then is the reverse lex leading such term, in any of Sw(x), Sw(x; a), code S˜ w(x),andS˜ w(x; a), is the monomial x (w),wherecode(w)=(c1,c2,...) and

ci = |{j ∈ Z>0 | i0.

Lemma 3.2 ([Mac]). There is a bijection from S∞ to the set of tuples (c1,c2,...) of nonnegative integers, almost all zero, given by w → code(w). Moreover it restricts to a bijection from Sn to the set of tuples (c1,...,cn) such that 0 ≤ ci ≤ n − i for all 0 ≤ i ≤ n.

Lemma 3.3. Let x =(x1,...,xn), a =(a1,...,an),andq =(q1,...,qn−1). Then:

(1) {Sw(x) | w ∈ Sn} is a Z-basis of Z[x]/Jn. { | ∈ } Z Z a (2) Sw(x; a) w Sn is a [a]-basis of [x; a]/Jn. { ˜ | ∈ } Z Z q (3) Sw(x) w Sn is a [q]-basis of [x; q]/Jn. { ˜ | ∈ } Z Z q,a (4) Sw(x; a) w Sn is a [q, a]-basis of [x; q; a]/Jn . Proof. Since in each case the highest degree part of the j-th ideal generator in the x variables is ej (x1,...,xn), any polynomial may be reduced modulo the ideal γ ∈ Zn until its leading term in the x variables is x ,whereγ =(γ1,...,γn) ≥0 with γi ≤ n − i for 1 ≤ i ≤ n. But these are the leading terms of the various kinds of Schubert polynomials. 3.2. Geometric bases. Under the isomorphism (2.1) (resp. (2.2), (2.3)) the Schu- w bert basis [Xw](resp.[Xw]T , σ ) corresponds to Sw(x)(resp. Sw(x; a), S˜ w(x)) by [LS, BGG] for H∗(Fl), [Bi] for HT (Fl), and [FGP] for QH∗(Fl). Our ﬁrst main result is: Theorem 3.4. Under the S[q]-algebra isomorphism (2.4) the quantum equivariant w Schubert basis element σT corresponds to the quantum double Schubert polynomial S˜ w(x; a). Theorem 3.4 is proved in Section 5.

1This is not the standard deﬁnition of a double Schubert polynomial. However it is easily seen − − to be equivalent using, say, the identity Sw−1 (x; a)=Sw( a; x)[Mac].

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r 3.3. Stable quantization. This section follows [FGP]. Let ei = ei(x1,x2,...,xr) be the elementary symmetric polynomial for integers 0 ≤i ≤ r.By[FGP, Prop. 3.3], Z[x]hasaZ-basis of standard monomials e = er ,whereI = I r≥1 ir (i1,i2,...) is a sequence of nonnegative integers, almost all zero, with 0 ≤ ir ≤ r for all r ≥ 1. The stable quantization map is the Z[q]-module automorphism θ of Z[x; q]givenby e → E := Ej . I I ij j≥1 By [FGP, CF, KM] we have2 ˜ (3.3) θ(Sw(x)) = Sw(x) for all w ∈ S∞. The map θ is extended by Z[a]-linearity to a Z[q, a]-module automorphism of Z[x, q, a]. 3.4. Cauchy formulae. The double Schubert polynomials satisfy [Mac] (3.4) Sw(x; a)= Svw−1 (−a)Sv(x), v≤Lw where v ≤L w denotes the left weak order, defined by (wv−1)+(v)=(w). For a geometric explanation of this identity see [And]. We have [KM, CF] ˜ ˜ (3.5) θ(Sw(x; a)) = Sw(x; a)= Svw−1 (−a)Sv(x). v≤Lw ˜ The first equality follows from the divided difference definitions of Sw(x; a)and Sw(x; a) and the commutation of the divided differences in the a-variables with quantization. The second equality follows from quantizing (3.4). We require the explicit formulae for Schubert polynomials indexed by simple reflections. Lemma 3.5. We have x ··· (3.6) Ssi (x)=ωi = x1 + x2 + + xi, ˜ (3.7) Ssi (x)=Ssi (x), ˜ x − a (3.8) Ssi (x; a)=ωi ωi .

Proof. Since si is an i-Grassmannian permutation with associated partition consist-

ing of a single box, its Schubert polynomial is the Schur polynomial [Mac] Ssi (x)= x ˜ S1[x1,...,xi]=ωi , proving (3.6). For (3.7) we have Ssi (x)=θ(Ssi (x)) = x ˜ ˜ − θ(e1,i)=E1,i = ωi . For (3.8), by (3.5) we have Ssi (x; a)=Ssi (x) Ssi (a)= x − a ωi ωi , as required.

4. Chevalley-Monk rules for Schubert polynomials The Chevalley-Monk formula describes the product of a divisor class and an arbitrary Schubert class in the cohomology ring H∗(Fl). The goal of this section is to establish the Chevalley-Monk rule for quantum double Schubert polynomials. The Chevalley-Monk rules for (double) Schubert polynomials should be viewed as

2This is the deﬁnition of a quantum Schubert polynomial in [FGP].

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product rules for the (equivariant) cohomology of an inﬁnite-dimensional ﬂag ind- variety Fl∞ of type A∞ with Dynkin node set Z>0 and simple bonds between i and i +1foralli ∈ Z>0. + { − | ≤ } ∨ − Let Φ = αij = ai aj 1 i

Proposition 4.2 ([Mac]). For w ∈ S∞ and i ∈ Z>0 the Schubert polynomials satisfy the identity in Z[x] given by ∨ (4.1) Ssi (x)Sw(x)= α ,ωi Swsα (x).

α∈Aw Example 4.3. It is necessary to take a large-rank limit (n 0) to compare 2 ∗ ∅ Propositions 4.1 and 4.2. Let n =2.Wehave[Xs1 ] =0inH (Fl2)sinceAs1 = 2 2 { } for SL2. Lifting to polynomials we have Ss = x1 = Ss2s1 since As1 = α13 , 1 ∈ ∈ \ which is not a positive root for SL2.NotethatSs2s1 J2 and s2s1 S3 S2.In ∗ ≥ 2 H (Fln)forn 3wehave[Xs1 ] =[Xs2s1 ]. 4.2. Quantum Schubert polynomials. Proposition 4.4 ([FGP]). For w ∈ S and 1 ≤ i ≤ n − 1,inQH∗(Fl ) we have n n si w ∨ wsα ∨ wsα σ σ = α ,ωiσ + qα∨ α ,ωiσ . ∈ n ∈ n α Aw α Bw

Proposition 4.5 ([FGP]). For i ∈ Z>0 and w ∈ S∞ the quantum Schubert polynomials satisfy the identity in Z[x; q] given by ∨ ∨ ˜ ˜ ˜ ∨ ˜ (4.2) Ssi (x)Sw(x)= α ,ωi Swsα (x)+ qα α ,ωi Swsα (x).

α∈Aw α∈Bw 4.3. Double Schubert polynomials. T Proposition 4.6 ([KK], [Rob]). For w ∈ Sn and 1 ≤ i ≤ n − 1,inH (Fln) we have − a · a ∨ (4.3) [Xsi ]T [Xw]T =( ωi + w ωi )[Xw]T + α ,ωi [Xwsα ]T . ∈ n α Aw The following result is a consequence of [Man, Ex. 2.7.3].

Proposition 4.7. For w ∈ S∞ and i ∈ Z>0, the double Schubert polynomials satisfy the identity in Z[x, a] given by − a · a ∨ (4.4) Ssi (x; a)Sw(x; a)=( ωi + w ωi )Sw(x; a)+ α ,ωi Swsα (x; a).

α∈Aw

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Proof. Fix w ∈ S∞. We observe that the set Aw is ﬁnite. Let N be large enough so that all appearing terms make sense for SN . By [Bi] under the isomorphism (2.2), → a ∈ T [Xw]T Sw(x; a)+JN for w SN . By Proposition 4.6 for H(FlN ), equation ∈ a (4.4) holds modulo an element f JN . We may write f = v∈S∞ bvSv(x; a) where bv ∈ Z[a] and only ﬁnitely many are nonzero. Choose n ≥ N large enough so that v ∈ Sn and bv ∈ Z[a1,...,an] for all v with bv = 0. Applying Proposition T ∈ a 4.6 again for H (Fln) we deduce that f Jn. By Lemma 3.3 it follows that f =0, as required. 4.4. Quantum double Schubert polynomials. Theorem 2.1 gives the equivari- T ant quantum Chevalley-Monk rule for QH (Fln). We cannot use the multiplication rule in Theorem 2.1 directly because we are trying to prove that the quantum double Schubert polynomials represent Schubert classes. We deduce the following product formula by cancelling down to the equivariant case which was proven above. Proposition 4.8. The quantum double Schubert polynomials satisfy the equivariant quantum Chevalley-Monk rule in Z[x, q, a]: for all w ∈ S∞ and i ≥ 1 we have ˜ ˜ − a · a ˜ ∨ ˜ (4.5) Ssi (x; a)Sw(x; a)=( ωi + w ωi )Sw(x; a)+ α ,ωi Swsα (x; a) ∈ α Aw ∨ ∨ ˜ + qα α ,ωi Swsα (x; a). ∈ α Bw Proof. Let C = α∨ ,ωS (x; a). Starting with (4.4) and using Lem- α∈Aw i wsα ma 3.5 and (3.4) we have − − a · a 0= Ssi (x; a)Sw(x; a)+( ωi + w ωi )Sw(x; a)+C =(ωa − S (x))S (x; a)+(−ωa + w · ωa)S (x; a)+C i si w i i w a − −1 − · = Ssi (x) Svw ( a)Sv(x)+(w ωi )Sw(x; a)+C ≤L v w ∨ a − −1 − · = Svw ( a) α ,ωi Svsα (x)+(w ωi )Sw(x; a)+C.

v≤Lw α∈Av Quantizing and rearranging, we have · a ˜ ∨ ˜ (w ωi )Sw(x; a)+ α ,ωi Swsα (x; a) ∈ αAw ∨ −1 − ˜ = Svw ( a) α ,ωi Svsα (x) ≤L ∈ v w α Av ∨ −1 − ˜ ˜ − ˜ = Svw ( a) Ssi (x)Sv(x) α ,ωi Svsα (x) ≤L ∈ v w α Bv ∨ ˜ ˜ − −1 − ˜ = Ssi (x)Sw(x; a) Svw ( a) α ,ωi Svsα (x).

v≤Lw α∈Bv Therefore to prove (4.5) it suﬃces to show that ∨ −1 − ∨ ˜ (4.6) Svw ( a) qα α ,ωi Svsα (x) ≤L ∈ v w α Bv ∨ ∨ −1 − ˜ = α ,ωi qα Svsαw ( a)Sv(x). ∈ L α Bw v≤ wsα

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To prove (4.6) it suﬃces to show that there is a bijection A → B given by (v, α) → (vsα,α), where + L A = {(v, α) ∈ W × Φ | v ≤ w and α ∈ Bv}, + L B = {(u, α) ∈ W × Φ | u ≤ wsα and α ∈ Bw}. Let (v, α) ∈ A.Thenw =(wv−1)(v) is length-additive since v ≤L w and v = ∨ (vsα)(sα) is length-additive because (sα)=α , 2ρ−1andα ∈ Bv. Therefore −1 −1 w =(wv )(vsα)(sα) is length-additive. It follows that wsα =(wv )(vsα)is L −1 length-additive and that vsα ≤ wsα.Moreoverwehave(wsα)=(wv )+ −1 ∨ ∨ (vsα)=(wv )+(v)+1−α , 2ρ = (w)+1−α , 2ρ. Therefore (vsα,α) ∈ B. Conversely suppose (u, α) ∈ B.Letv = usα. Arguing as before, we have that −1 −1 L w =(wsαu )(u)(sα)=(wv )(v) is length-additive. We deduce that v ≤ w and −1 ∨ −1 ∨ that (v)=(w)−(wv )=(wsα)+1−α , 2ρ−(wv )=(vsα)+1−α , 2ρ so that α ∈ Bv, as required.

5. Proof of Theorem 3.4 a Z p − p ≥ ≥ Let I be the ideal in [x, a] generated by ei (x) ei (a)forp n and i 1, a and ai for i>n.LetJ ⊂ Z[x, a]betheZ[a]-submodule spanned by Sw(x; a) for w ∈ S∞ \ Sn and aiSu(x; a)fori>nand any u ∈ S∞.Weshallshowthat a a I = J .Letci,p = sp+1−i ···sp−2sp−1sp ∈ S∞ \ Sp be the cycle of length i.We { p − p | ≤ ≤ } Z note that the family ei (x) ei (a) 1 i p is unitriangular over [a]withthe { | ≤ ≤ } Z Z family Sci,p (x; a) 1 i p .Since [x, a]= u∈S∞ [a]Su(x; a), to show that a ⊂ a ∈ a ≥ ≥ I J it suffices to show that Su(x; a)Sci,p (x, a) J for all p n, i 1, and u ∈ S∞. But this follows from the fact that the product of Su(x, a)Sv(x; a)isa Z[a]-linear combination of Sw(x; a)wherew ≥ u and w ≥ v. a Let K be the ideal in Z[x, a] generated by ai for i>n.ThenI /K has a Z[a1,...,an]-basis given by standard monomials eI with ir > 0forsomer ≥ n, a while J /K has a Z[a1,...,an]-basis given by Sw(x; a)forw ∈ S∞ \ Sn.The quotient ring Z[x, a]/K has a Z[a1,...,an]-basis given by all standard monomials eI for I =(i1,i2,...)with0≤ ip ≤ p for all p ≥ 1 and almost all ip zero, and also by all double Schubert polynomials Sw(x; a)forw ∈ S∞. But the standard monomials eI with in = in+1 = ··· = 0 are in graded bijection with the Sw(x; a) a a for w ∈ Sn. It follows that I = J by graded dimension counting. qa Z p − p ≥ ≥ Let J∞ be the ideal of [x, q, a] generated by Ei ei (a) for all i 1andp n, together with qi for i ≥ n and ai for i>n. We wish to show that qa (5.1) S˜ w(x; a) ∈ J∞ for all w ∈ S∞ \ Sn. For this it suffices to show that a qa (5.2) θ(J∞) ⊂ J∞ . To prove (5.2) it suffices to show that p − p ∈ qa ≥ ≥ (5.3) θ(eI (ei (x) ei (a))) J∞ for standard monomials eI , i 1andp n. p To apply θ to this element we must express eI ei in standard monomials. The only nonstandardness that can occur is if ip > 0. In that case one may use [FGP, (3.2)]: p p p p p p+1 − p p+1 ei ej = ei−1ej+1 + ej ei ei−1ej+1. p Ultimately, the straightening of eI ei into standard monomials only changes factors q ≥ ≥ of the form ek for k 1andq p.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUANTUM DOUBLE SCHUBERT POLYNOMIALS 843 Let E = Er for I =(i ,i ,...). If we consider E (Ep − ep(a)) and use I r≥1 ir 1 2 I i i [FGP, (3.6)] p p p p p p+1 − p p+1 p−1 p − p−1 p Ei Ej = Ei−1Ej+1 + Ej Ei Ei−1Ej+1 + qp(Ej−1 Ei−1 Ei−2 Ej ) to rewrite it into quantized standard monomials, we see that the two straightening processes differ only by multiples of qp, qp+1, etc. Therefore p − p − p − p ∈ qa θ(eI (ei (x) ei (a))) EI (Ei ei (a)) J∞ . p − p ∈ qa But EI (Ei ei (a)) J∞ , so (5.3) holds and (5.1) follows. qa The ring Z[x, q, a]/J∞ has a Z[q1,...,qn−1; a1,...,an]-basis given by S˜ w(x; a) for w ∈ Sn. This follows from Lemmata 3.1 and 3.2. Moreover this basis satisfies the equivariant quantum Chevalley-Monk rule for SLn by Proposition 4.8. By Theorem 2.1 there is an isomorphism of Z[q1,...,qn−1; a1,...,an]-algebras T → Z qa w ˜ QH (SLn/B) [x, q, a]/J∞ .Moreover,σT and Sw(x; a) (or, rather, its pre- image in QHT (Fl)) are related by an automorphism of QHT (Fl). But the Schubert si divisor class σT is (by definition) represented by a usual double Schubert polyno- ˜ mial Ssi (x; a)=Ssi (x; a) (Lemma 3.5) in Kim’s presentation, and these divisor T classes generate QH (Fl) over Z[q1,...,qn−1; a1,...,an]. Thus the automorphism must be the identity, completing the proof.

6. Parabolic case ∈ Zk ··· 6.1. Notation. Fix a composition (n1,n2,...,nk) >0 with n1 +n2 + +nk = n.LetP ⊂ SLn(C) be the parabolic subgroup consisting of block upper triangular matrices with block sizes n1,n2,...,nk.ThenSLn/P is isomorphic to the variety n of partial ﬂags in C with subspaces of dimensions Nj := n1 + n2 + ···+ nj for P 0 ≤ j ≤ k.DenotebyWP the Weyl group for the Levi factor of P and by W the set of minimum length coset representatives in W/WP . For every w ∈ W there P P P exist unique elements w ∈ W and wP ∈ WP such that w = w wP ;moreover this factorization is length-additive. Let w0 ∈ W be the longest element and let P ∈ w0 = w0 w0,P so that w0,P WP is the longest element. P Example 6.1. Let (n1,n2,n3)=(2, 1, 3). Then (N1,N2,N3)=(2, 3, 6), w0 = 564123 (that is, w ∈ S6 is the permutation with w(1) = 5, w(2) = 6, etc.), and w0,P = 213654. 6.2. Parabolic quantum double Schubert polynomials. In this section let x =(x1,...,xn)andq =(q1,...,qk−1). Let Z[x; q] be the graded polynomial ring with deg(xi) = 1 and deg(qj)=nj + nj+1 for 1 ≤ j ≤ k − 1. Following [AS], P [Cio2] let D = D be the n × n matrix with entries xi on the diagonal, −1onthe nj+1 superdiagonal, and entry (Nj+1,Nj−1 +1)givenby −(−1) qj for 1 ≤ j ≤ k − 1. For 1 ≤ j ≤ k let Dj be the upper left Nj × Nj submatrix of D and for 1 ≤ i ≤ Nj j ∈ Z deﬁne the elements Gi [x; q]by Nj − − − Nj i j det(Dj t Id) = ( t) Gi . i=0 For w ∈ W P , we introduce the parabolic quantum double Schubert polynomial − k−1 nNj P −1 ˜ P − (w(w0 ) ) a − (6.1) Sw(x; a)=( 1) ∂ P −1 det(Dj aiId). w(w0 ) j=1 i=n−Nj+1+1

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Example 6.2. Continuing Example 6.1 we have ˜ P S P (x; a)=det(D1 − a4Id) det(D2 − a1Id) det(D2 − a2Id) det(D2 − a3Id) w0 3 =(x1 − a4)(x2 − a4) ((x1 − ai)(x2 − ai)(x3 − ai)+q1). i=1 Example 6.3. For the Grassmannian Gr(r, n)ofr-dimensional subspaces of Cn we have k =2,(n1,n2)=(r, n − r), (N0,N1,N2)=(0,r,n), and n−r P −1 ˜ P − (w(w0 ) ) a − Sw(x; a)=( 1) ∂ P −1 det(D1 aiId). w(w0 ) i=1

Here D1 is the r × r matrix with diagonal entries xi for 1 ≤ i ≤ r and superdiagonal entries −1; the variable q1 does not appear. One may verify that in this case the parabolic quantum double Schubert polynomial is just a double Schubert P polynomial Sw(x; a)forw ∈ W ; these are also known as double or factorial Schur polynomials. This agrees with [Mi2]. ˜ P The parabolic quantum double Schubert polynomials Sw(x; a) have specializa- tions similar to the quantum double Schubert polynomials. Let w ∈ W P . (1) We deﬁne the parabolic quantum Schubert polynomials by the specializa- ˜ P ˜ P tion Sw(x)=Sw(x;0) which sets ai =0foralli. In Lemma 6.9 it is shown that these polynomials coincide with those of Ciocan-Fontanine [Cio2], whose deﬁnition uses a parabolic analogue of the quantization map of [FGP]. (2) Setting qi =0foralli, one obtains the double Schubert polynomial Sw(x; a). (3) Setting both ai and qi to zero, one obtains the Schubert polynomial Sw(x). q qa Z WP a ⊂ WP ⊂ Z WP ⊂ Let JP be the ideal in [x] (resp. JP S[x] , JP [x] [q], JP WP n n − n k k − n S[x] [q]) generated by the elements ei (x)(resp. ei (x) ei (a), Gi , Gi ei (a)) for 1 ≤ i ≤ n. The aim of this section is to establish (4) of the following theorem. Theorem 6.4. (1) There is an isomorphism of Z-algebras [BGG, LS]:

∗ ∼ WP H (SLn/P ) = Z[x] /JP

[Xw] → Sw(x)+JP . (2) There is an isomorphism of S-algebras [Bi]:

T ∼ WP a H (SLn/P ) = S[x] /JP → a [Xw]T Sw(x; a)+JP . (3) There is an isomorphism of Z[q]-algebras [AS], [Kim2], [Kim3]: ∗ ∼ Z WP q QH (SLn/P ) = [x] [q]/JP P,w → ˜ P q σ Sw(x)+JP . (4) There is an isomorphism of S[q]-algebras :

T ∼ WP qa (6.2) QH (SLn/P ) = S[x] [q]/JP P,w → ˜ P qa (6.3) σT Sw(x; a)+JP .

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P,w P,w Here [Xw], [Xw]T , σ ,andσT denote the Schubert bases for their respective cohomology rings for w ∈ W P . The isomorphism (6.2) is due to [Kim3]. We shall establish (6.3), namely, that under this isomorphism, the cosets of parabolic quantum double Schubert polynomials are the images of parabolic equivariant quantum Schubert classes. Example 6.5. Continuing Example 6.3, the relations for QHT (Gr(r, n)) equate n−i ei(a) with the coeﬃcient of (−t) in det(D2 − tId). The n × n matrix D2,aside from the usual diagonal entries xi and superdiagonal entries −1, has only one other n−r nonzero entry −(−1) q1, which appears in position (n, 1). Therefore we have n−r relations ei(x) − ei(a)for1≤ i ≤ n − 1anden(x) − en(a) − (−1) q1. This agrees with [AS, §5.2]. 6.3. Stability of parabolic quantum double Schubert polynomials. The following lemma can be veriﬁed by direct computation and induction. ∈ Zn ≤ − ≤ ≤ Lemma 6.6. Let β =(β1,...,βn) ≥0 be such that βi n i for 2 i n. a ··· a a · β β2 β3 ··· βn − − Then ∂n−1 ∂2 ∂1 a equals a1 a2 an−1 if β1 = n 1 and 0 if β1

This means that if we append a block of size m to n• and append m ﬁxed points to w ∈ W P , the parabolic quantum double Schubert polynomial remains the same.

6.4. Stable parabolic quantization. This section follows [Cio2]. Consider an ··· inﬁnite sequence of positive integers n• =(n1,n2,...). Let Nj = n1 + + nj ≥ for j 1. Let W = S∞ = n≥1 Sn be the inﬁnite symmetric group, WP the P subgroup of W generated by si for i/∈{N1,N2,...}, W the set of minimum P length coset representatives in W/WP ,etc.LetY be the set of tuples of partitions

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• (1) (2) (j) λ =(λ ,λ ,...) such that λ is contained in the rectangle with nj+1 rows and (j) Nj columns and almost all λ are empty. Deﬁne nj+1 nj+1 j j • • gλ = g (j) ,Gλ = G (j) , λi λi j≥1 i=1 j≥1 i=1

j Nj where gr = er (x). The following is a consequence of [Cio2] by taking a limit.

• P WP Proposition 6.7 ([Cio2]). {gλ• | λ ∈ Y } is a Z-basis of Z[x] . We also observe that

P WP Lemma 6.8. {Sw(x) | w ∈ W } is a Z-basis of Z[x] . P Define the Z[q]-module automorphism θ of Z[x, q]byθ(gλ• )=Gλ• .ByZ[a]- linearity it defines a Z[q, a]-module automorphism of Z[x, q, a], also denoted θP . Lemma 6.9. For all w ∈ W P , ˜ P P (6.4) Sw(x)=θ (Sw(x)), ˜ P P (6.5) Sw(x; a)=θ (Sw(x; a)). ∈ P ˜ P Proof. Let w W . We may compute Sw(x; a) by working with respect to (n1,n2,...,nk) for some finite k; the result is independent of k by the previous subsection. Then (6.5) is an immediate consequence of the commutation of θP and the divided difference operators in the a variables. Equation (6.4) follows from (6.5) by setting the a variables to zero. 6.5. Cauchy formula. Keeping the notation of the previous subsection, we have Proposition 6.10. For all w ∈ W P we have P P ˜ − − ˜ (6.6) Sw(x; a)= Svw 1 ( a)Sv (x). v≤Lw ≤L ∈ P ˜ P Proof. Observe that if v w,thenv W so that Sv (x) makes sense. We have S˜ P (x; a)=θp(S (x; a)) w w P = θ ( Svw−1 (−a)Sv(x)) ≤L v w P − − ˜ = Svw 1 ( a)Sv (x). v≤Lw

6.6. Parabolic Chevalley-Monk rules. Fix n• =(n1,n2,...,nk) such that k ∨ + j=1 nj = n and let P be the parabolic deﬁned by n•,andsoon.LetQP ,ΦP ,and ρP be respectively the coroot lattice, the positive roots, and the half sum of positive ∨ → ∨ ∨ roots for the Levi factor of P .LetηP : Q Q /QP be the natural projection. P P P P P Let πP : W → W be the map w → w where w = w wP and w ∈ W and wP ∈ WP . Deﬁne the sets of roots n { ∈ + \ + | ∈ P } (6.7) AP,w = α Φ ΦP wsα w and wsα W , n { ∈ + \ + | − ∨ − } (6.8) BP,w = α Φ ΦP (πP (wsα)) = (w)+1 α , 2(ρ ρP ) .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUANTUM DOUBLE SCHUBERT POLYNOMIALS 847 k−1 ∨ ∨ ∨ b For λ = b α ∈ Q /Q with b ∈ Z≥ we let q = q i . The following is i=1 i Ni P i 0 λ i i T Mihalcea’s characterization of QH (SLn/P ) which extends Theorem 2.1.

P Theorem 6.11 ([Mi]). For w ∈ W and i ∈{N1,N2,...,Nk−1} (that is, si ∈ W P ), we have

σP,si σP,w =(−ωa + w · ωa)σP,w T T i i T ∨ P,wsα ∨ P,πP (wsα) ∨ + α ,ωi σT + α ,ωi qηP (α )σT . ∈ n ∈ n α AP,w α BP,w

{ P,w | ∈ P } Moreover these structure constants determine the Schubert basis σT w W T and the ring QH (SLn/P ) up to isomorphism as Z[q1,...,qk−1; a1,...,an]- algebras.

Now let (n1,n2,...) be an inﬁnite sequence of positive integers. Let AP,w and BP,w be the analogues of the sets of roots deﬁned in (6.7) and (6.8) where ρ = (0, −1, −2,...)andρP is the juxtaposition of (0, −1,...,1 − nj)forj ≥ 1.

P Proposition 6.12. For w ∈ W and i such that i ∈{N1,N2,...,} (that is, P si ∈ W ), the parabolic quantum double Schubert polynomials satisfy the identity ˜ P ˜ P − a · a ˜ P ∨ ˜ P Ssi (x; a)Sw(x; a)=( ωi + w ωi )Sw(x; a)+ α ,ωi Swsα (x; a) α∈A P,w ∨ P + α ,ωq ∨ S˜ (x; a). i ηP (α ) πP (wsα) α∈BP,w Sketch of the proof. The proof proceeds entirely analogously to that of Proposi- tion 4.8, starting with the Chevalley-Monk formula for double Schubert polynomi- P P als (applied in the special case that w ∈ W and si ∈ W ) and reducing to the following identity: ∨ P S −1 (−a) q ∨ α ,ωS˜ (x) vw ηP (α ) i πP (vsα) ≤L α∈B v w P,v ∨ P ∨ −1 − ˜ = α ,ωi qηP (α ) Sv(πP (wsα)) ( a)Sv (x). ∈ L α BP,w v≤ πP (wsα)

For this it suﬃces to show that there is a bijection (v, α) → (πP (vsα),α) and inverse bijection (u, α) → (πP (usα),α) between the sets { ∈ P × + \ + | ≤L ∈ } A = (v, α) W (Φ ΦP ) v w and α BP,v , { ∈ P × + \ + | ≤L ∈ } B = (u, α) W (Φ ΦP ) u πP (wsα)andα BP,w such that for (v, α) ∈ A we have

−1 −1 (6.9) vw = πP (vsα)(πP (wsα)) .

To establish this bijection we use [LamSh, Lemma 10.14], which asserts that elements α ∈ BP,w automatically satisfy the additional condition (wsα)=(w)+ 1 −α∨ , 2ρ.Let(v, α) ∈ A. Then we have length-additive factorizations w = −1 (wv ) · v, v =(vsα) · sα,andvsα = πP (vsα) · x for some x ∈ WP . Therefore

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−1 −1 w =(wv ) · πP (vsα) · x · sα is length-additive. This implies that wsα =(wv ) · πP (vsα) · x is length-additive. One may deduce from this that πP (wsα)= −1 (wv )πP (vsα) and therefore that (6.9) holds. The conditions that (πP (vsα),α)∈B are readily veriﬁed. Thus the map A → B is well-deﬁned. The rest of the proof is similar.

6.7. Proof of Theorem 6.4(4). The proof is analogous to that of Theorem 3.4. Given the finite sequence (n1,...,nk) and parabolic subgroup P , consider the extension (n ,...,n , 1, 1, 1,...) by an infinite sequence of 1s. We use the notation 1 k ∼ P∞ to label the corresponding objects. That is, WP = WP∞ is the subgroup of S∞, P∞ W is the set of minimum-length coset representatives in S∞/WP∞ ,andsoon. q Z WP∞ Z WP∞ Let RP = [x, a] and RP = [x, q, a] ,wherex =(x1,x2,...), q = (q1,q2,...), and q1,q2,...,qk−1 are identified with the quantum parameters in The- a p − Np orem 6.4(4). Define JP,∞ to be the ideal of RP generated by gi (x) ei (a)for i ≥ 1andp ≥ k and by ai for i>n. Note that for p = k + j with j ≥ 0wehave p − Np n+j − n+j gi (x) ei (a)=ei (x) ei (a). a Z ∈ P∞ \ Let IP,∞ be the [a]-submodule of RP spanned by Sw(x; a)forw W Sn, ∈ P∞ a and by aiSu(x; a)fori>nand any u W . We claim that IP,∞ is an ideal. This follows easily from the fact that the only double Schubert polynomials occurring in the expansion of a product Sw(x; a)Sv(x; a) lie above w and v in Bruhat order. a a But then it follows from Theorem 6.4(2) and dimension counting that IP,∞ = JP,∞. qa q p − Np ≥ ≥ Define JP,∞ to be the ideal of RP generated by Gi ei (a)forp k and i 1, ≥ ˜ P ∈ qa ∈ P∞ \ and ai for i>n,andqi for i k. We claim that Sw(x; a) JP,∞ for w W Sn. P∞ a ⊂ This would follow from the definitions if we could establish that θ (JP,∞) qa P∞ JP,∞.Sinceθ is trivial on the equivariant variables a1,a2,..., it suffices to show P∞ P p − Np ∈ qa ≥ ≥ that θ (gλ• (gi (x) ei (a))) JP,∞,foreachi 1, p k, and each standard P P∞ P p monomial gλ• . To apply θ to gλ• gi we must first standardize this monomial. This can be achieved by using a parabolic version of the straightening algorithm of P p [FGP, Proposition 3.3]. The only nonstandard part of gλ• gi is the possible presence p p of a factor gj gi . This can be standardised using only the (non-parabolic) relation k P p [FGP, (3.2)] – any factors gm for k

− − GpGp+1 + Gp Gp ± q Gp 1 Gp = GpGp+1 + Gp Gp ± q Gp 1 Gp. i j+1 i+1 j p i−np+1−np j j i+1 k+1 i p j−np+1−np i

We note that when p ≥ k, this relation only involves quantum variables qk,qk+1,.... P p P p Thus modulo qk,qk+1,..., the straightening relation for gλ• gi and for Gλ• Gi co- P∞ P p − Np P p − Np qa incide. It follows that θ (gλ• (gi (x) ei (a))) = Gλ• (Gi ei (a)) mod JP,∞, ˜ P ∈ qa ∈ P∞ \ qa qa | P | and thus Sw(x; a) JP,∞ for w W Sn.ButRP,∞/JP,∞ has rank W over qa ˜ P ∈ Z[q1,...,qk−1,a1,...,an], and so it follows that JP,∞ is spanned by Sw(x; a) qa ∈ P∞ \ ˜ P ≥ ˜ P JP,∞ for w W Sn together with qiSu (x; a)fori k, aiSu (x; a)fori>n and any u ∈ W P∞ . Theorem 6.4(4) now follows from Proposition 6.12 and the determination of T QH (SLn/P ) in Theorem 6.11.

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Acknowledgement The authors thank Linda Chen for communicating to them her joint work with Dave Anderson [AC].

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