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 polynomi- als 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 Sch¨utzenberger [LS]; the double Schubert polynomials, also due to Lascoux and Sch¨utzenberger, 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 poly- nomials 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 836 THOMAS LAM AND MARK SHIMOZONO Let SLn/P be a partial flag variety, where P denotes a parabolic subgroup of SLn. In nonquantum Schubert calculus, the functorality of (equivariant) cohomol- ogy 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 quan- tum 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use QUANTUM DOUBLE SCHUBERT POLYNOMIALS 837 ∗ 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 838 THOMAS LAM AND MARK SHIMOZONO (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.