Quantum Cohomology of Grassmannians

Compositio Mathematica 137: 227–235, 2003. 227 # 2003 Kluwer Academic Publishers. Printed in the Netherlands.

ANDERS SKOVSTED BUCH? Matematish Institut, Aarhus Universitet, Ny Munkegade, 8000 A˚rhus C, Denmark. e-mail: [email protected]

(Received: 18 October 2001; accepted: 14 March 2002)

Abstract. We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about theminimal q-power which appears in a product of two Schubert classes. Mathematics Subject Classiﬁcations (2000). 14N35 (primary); 14M15, 05E05 (secondary) Key words. quantum cohomology, Gromov–Witten invariants, Grassmann varieties.

1. Introduction The purpose of this paper is to give simple proofs of the main theorems about the (small) quantum cohomology ring of a Grassmann variety. This ﬁrst of all includes Bertram’s quantum versions of the Pieri and Giambelli formulas [1]. Bertram’s proofs of these theorems required the use of quot schemes. Our proof of the quantum Pieri formula uses no moduli spaces and only the deﬁnition of Gromov–Witten invariants. In fact we show that this formula is a consequence of the classical Pieri formula. We then show that the quantum Giambelli formula follows immediately from the quantum Pieri formula together with the classical Giambelli formula and associativity of quantum cohomology [13, 17]. Wealso givea short proof of theGrassmannian caseof a formula of Fulton and Woodward for theminimal q-power which appears in a quantum product of two Schubert classes [10]. In addition we supply a proof of a simple version of the rim- hook algorithm [2] based on ‘mod n’ arithmetic which is due to F. Sottile [19]. Finally we recover the presentation of the quantum cohomology of Grassmannians [18, 21]. In this paper we only assume associativity of quantum cohomology and standard facts about theusual cohomology. Thebasic ideais that if a rational curve of degree d passes through a Schubert variety in the Grassmannian Grðl; CnÞ of l- dimensional subspaces of Cn, then the linear span of the points of this curve gives riseto a point in Gr ðl þ d; CnÞ which must lie in a related Schubert variety. Remark- ably, this simple idea can in many cases be used to conclude that no curves pass through three Schubert varieties in general position because the intersection of the

?Theauthor was partially supportedby NSF Grant DMS-0070479.

related Schubert varieties in Grðl þ d; CnÞ is empty. In particular, the quantum Giambelli formula can be deduced by knowing that certain Gromov–Witten invariants are zero, and in each case this follows because the codimensions of the related Schubert varieties add up to more than the dimension of Grðl þ d; CnÞ. The same idea can be used to prove the quantum Monk’s formula for ﬂag varieties SLn=B [4, 7] and even the quantum Pieri formula for partial ﬂag varieties SLn=P [5, 6]. In a paper [3] with A. Kresch and H. Tamvakis this idea will furthermore be applied to obtain a similar treatment of the quantum cohomology of Lagrangian and orthogonal Grassmannians [14, 15].

2. Preliminaries n Set E ¼ C , X ¼ Grðl; E Þ, and k ¼ n À l. Given a ﬂag of subspaces F1 F2 ÁÁÁFn ¼ E and a partition l ¼ðl1 5l2 5 ÁÁÁ 5ll 5 0Þ with l1 4 k, wedeﬁne the Schubert variety

OlðFÞ¼fV 2 X j dimðV \ FkþiÀl Þ 5 i 81 4 i 4 lg: ð1Þ i P The codimension of this variety is equal to the weight jlj¼ li of l. Welet Ol Ã Ã denotetheclass of OlðFÞ in thecohomology ring H X ¼ H ðX; ZÞ. TheSchubert classes Ol form a basis for this ring. The partitions indexing this basis are exactly thosewheretheYoung diagram ﬁts in an l Â k rectangle. Ã The cohomology ring has the presentation H X ¼ Z½O1; ...; Ok=ðYlþ1; ...; YnÞ where Yp ¼ detðO1þjÀiÞ14i; j4p. Here we set Oi ¼ 0 for i < 0ori > k for convenience. In this presentation the class Ol is given by the Giambelli formula

Ol ¼ detðOliþjÀiÞ14i; j4l: ð2Þ This is usually deduced (cf. [8]) from the Pieri formula, which states that X Oi Á Ol ¼ On ð3Þ where the sum is over all partitions n which can beobtainedby adding i boxes to the Young diagram of l with no two in thesamecolumn. Recall that the degree of a rational curve f: P1 ! PN is equal to the number of points in theinverseimageby f of a general hyperplane in PN. The degree of a map 1 f: P !VX is the degree of the composition of f with thePlu ¨ cker embedding l X Pð E Þ which maps a point V 2 X to v1 ^ÁÁÁ^vl for any basis fv1; ...; vlg of V. Now let l, m, and n be three partitions contained in an l Â k rectangle, and let d 5 0 be an integer such that jljþjmjþjnj¼lk þ dn. TheGromov–Witteninvariant

hOl; Om; Onid is defined as the number of rational curves of degree d on X, which meet all of the Schubert varieties OlðFÞ, OmðGÞ,andOnðHÞ for general flags F, 1 G, H, up to automorphisms of P . A simple proof that this number is well defined is given in [1]. If jljþjmjþjnj 6¼ lk þ dn; then hOl; Om; Onid ¼ 0. The(small) quantum cohomology ring QHÃX ¼ QHÃðX; ZÞ of X is a Z½q-algebra Ã which is isomorphic to H X Z Z½q as a moduleover Z½q. In this ring wehave Ã Schubert classes sl ¼ Ol 1. Thering structureon QH X is defined by

Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 27 Sep 2021 at 21:33:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1023908007545 QUANTUM COHOMOLOGY OF GRASSMANNIANS 229 X d sl Á sm ¼ hOl; Om; On_ id q sn ð4Þ n;d50 _ where n ¼ðk À nl; k À nlÀ1; ...; k À n1Þ is thepartition for thedual Schubertclass of On. It is a nontrivial fact that this defines an associative ring structure [13, 17] (see Ã also [9]). Notice that the definition implies that QH X is a graded ring where sl has degree jlj and q has degree n. Furthermore, the map QHÃX=ðqÞ!HÃX which sends sl to Ol is an isomorphism of rings, so thequantum ring is a deformation of theusual cohomology ring. Siebert and Tian have given a presentation of this ring which is similar to the above presentation of the cohomology ring [18]. We will comment on this presentation towards the end of this paper. For now we will start from the basic assumption that the quantum ring is a well defined associative ring, and aim towards proving generalizations of the Pieri and Giambelli formulas stated above to reveal its structure.

3. The Span and Kernel of a Curve Our main new tool is the following definition. If Y is any subvariety of X ¼ Grðl; E Þ wedefinethe span of Y to bethesmallestsubspaceof E containing all the l-dimensional spaces given by points of Y. Similarly wedefinethe kernel of Y to bethelar- gest subspace of E contained in all the spaces given by points of Y.

LEMMA 1. Let C be a rational curve of degree d in X. Then the span of C has dimension at most l þ d and the kernel of C has dimension at least l À d. 1 Proof. Let C betheimageof a regular function f: P ! X ofL degree d, and let Ã l S E OX bethetautological subbundleon X. ThenLf S ¼ i¼1 OP1 ðÀaiÞ for l integers ai 5 0 with sum d, and f is given by an inclusion i¼1 OP1 ðÀaiÞE OP1 , i.e. a point p 2 P1 is mapped to the ﬁber over p of theimageof this bundlemap. If 1 ðs : tÞ are homogeneous coordinates on P then GðOP1 ðaiÞÞ has thebasis j aiÀj 1 1 fs t g04j4ai , so each map OP ðÀaiÞ!E OP has theform

Xai Xai Àj jÀai ðiÞ aj s t 7! vj aj j¼0 j¼0

ðiÞ Ã forL vectors vj 2 E (which will depend on the chosen identiﬁcation of f S with l i¼1 OP1 ðÀaiÞ ). Thespan ofPC must therefore be contained in the span of the set ðiÞ fvj g which has cardinality ð1 þ aiÞ¼l þ d. On the other hand, at least l À d of the integers ai must be zero, and the kernel of C contains thespan of thecorrespond- ðiÞ ing vectors v0 . &

The above lemma can also be obtained by proving that any regular function 1 1 f: P ! X can bewrittenas fðtÞ¼f1ðtÞ^ÁÁÁ^flðtÞ for maps fi : P ! PðE Þ. This is what we did until A. Kresch showed us the simpler argument given here.

If l is a partition and d a nonnegative integer, we let l^ denote the partition obtained by removing the leftmost d columns from theYoung diagram of l, i.e. ^ li ¼ maxðli À d; 0Þ.

LEMMA 2. Let C X be a rational curve of degree d 4 k and let W Ebeanlþ d dimensional subspace containing the span of C. If l is a partition such that

C \ OlðFÞ 6¼;then W belongs to the Schubert variety Ol^ ðFÞ in Grðl þ d; E Þ. Proof. Let V 2 C \ OlðFÞ. Since V W, theSchubertconditions on V imply ð \ that dim W FkþiÀli Þ 5 i for all i, which says exactly that W belongs to Ol^ ðFÞ. &

Similarly, if K E is an l À d dimensional subspace contained in the kernel of C, then K 2 Ol ðFÞGrðl À d; E Þ, where l ¼ðldþ1; ...; llÞ is the result of removing the top d rows of l. Therefore, the condition that a curve meets a given set of Schubert varieties in X implies that intersections of related Schubert varieties in Grðl þ d; E Þ and in Grðl À d; E Þ are not empty, which is a statement about the usual cohomology of these spaces. As we shall see, this simple idea is sufﬁcient to compute Gromov– Witten invariants in many important cases.

4. The Quantum Pieri Formula Westart with thefollowing quantum version of thePieriformula [1].

THEOREM 1 (Bertram). If l is contained in an l Â k rectangle and p 4 k then X X sp Á sl ¼ sm þ q sn

where the ﬁrst sum is over all partitions m such that jmj¼jljþp and k 5m1 5 l1 5m2 5l2 5 ÁÁÁ 5ml 5ll, and the second sum is over all partitions n such that jnj¼jljþp À n and l1 À 1 5n1 5l2 À 1 5n2 5 ÁÁÁ 5ll À 1 5nl 5 0.

Recall that the length ‘ðlÞ of a partition l is thenumberof non-zeroparts of l. Notice that the second sum in the theorem is nonzero only if ‘ðlÞ¼l.

Proof. The ﬁrst sum is dictated by the classical Pieri formula. Notice that this classical case is equivalent to the following statement. If a and b arepartitions such that jajþjbjþp ¼ lk then n 1ifa þ b 5 k for i þ j ¼ l and a þ b 4 k for i þ j ¼ l þ 1; hO ; O ; O i ¼ i j i j a b p 0 0 otherwise: Now suppose jajþjbjþp ¼ lk þ dn for some d 5 1 and let C bea rational curve of degree d in X which meets each of the varieties OaðFÞ, ObðGÞ,andOpðHÞ for general ﬂags F, G, H. Noticethat dn 4 lk þ k which implies that d 4 k.Let W E bea subspaceof dimension l þ d which contains thespan of C. Then ^ W 2 Grðl þ d; E Þ lies in the intersection Oa^ ðFÞ\Ob^ ðGÞ\Op^ðHÞ where a^ and b are the results of removing the leftmost d columns from a and b, and

p^ ¼ maxð p À d; 0Þ. Sincetheﬂags F, G, H are general, this implies that ja^jþjb^jþp^ 4 ðl þ d Þðk À d Þ. Sincewealso have

ja^jþjb^jþp^ 5 jajþjbjÀ2ld þ p À d ¼ðl þ d Þðk À d Þþd 2 À d wededucethat d ¼ 1 and ‘ðaÞ¼‘ðbÞ¼l. Using this, thequantum Pieriformula becomes equivalent to the statement that if jajþjbjþp ¼ lk þ n then

hOa; Ob; Opi1 ¼ 1ifai þ bj 5 k þ 1 for i þ j ¼ l þ 1 and ai þ bj 4 k þ 1 for i þ j ¼ l þ 2; otherwise hOa; Ob; Opi1 ¼ 0. In other words, the quantum Pieri formula

states that hOa; Ob; Opi1 ¼hOa^ ; Ob^ ; Op^i0 where the right hand side is a coefﬁcient of theclassical Pieriformula for Gr ðl þ 1; E Þ.

If hOa^ ; Ob^ ; Opî0 is zero then the space W can’t exist, so neither can C. On theother hand, if hOa^ ; Ob^ ; Opî0 ¼ 1 then there exists a unique subspace W E of dimension l þ 1 which is contained in the intersection Oa^ ðFÞ\Ob^ ðGÞ\Op^ðHÞ. Furthermore, sincetheflags aregeneral, W must lie in the interior of each of these Schubert vari-

eties. In particular, each of the spaces V1 ¼ W \ FnÀal and V2 ¼ W \ GnÀbl have dimension l. Noticealso that V1 2 OaðFÞ and V2 2 ObðGÞ. Since OaðFÞ\ ObðGÞ¼;wededucethat V1 6¼ V2,soK ¼ V1 \ V2 has dimension l À 1. Wecon- clude that the only rational curve of degree 1 in X which meets the Schubert varieties for a, b,andp is theline PðW=K Þ of l-dimensional subspaces between K and W. &

5. The Quantum Giambelli Formula

Now set si ¼ 0 for i < 0 and for k < i < n. The quantum Giambelli formula states that the classical formula (2) continues to be valid if all Schubert classes are replaced with the corresponding quantum Schubert classes [1].

THEOREM 2 (Bertram). If l is a partition contained in an l Â k rectangle, then the Ã Schubert class sl in QH Grðl; E Þ is given by sl ¼ detðsliþjÀiÞ14i; j4l.

Proof. Weclaim that if 0 4 ij 4 k for 1 4 j 4 l then si1 Á si2 ÁÁÁsil ¼

ðOi1 Á Oi2 ÁÁÁOil Þ 1, i.e. no q-terms show up when the ﬁrst product is expanded in the quantum ring. Using induction, this can be established by proving that if ‘ðmÞ < l then the expansion of si Á sm involves no q-terms and no partitions of lengths greater than ‘ðmÞþ1. The claim therefore follows from Theorem 1. Since the determinant of the quantum Giambelli formula is a signed sum of products of the form

si1 Á si2 ÁÁÁsil , weconcludefrom theclassical Giambelliformula (2) that det ðsliþjÀiÞ¼

detðOliþjÀiÞ 1 ¼ Ol 1 ¼ sl as required. &

Noticethat this proof usesonly that no q-terms are contained in a product si Á sm when ‘ðmÞ < l (in addition to the classical Giambelli and Pieri formulas). In fact, Theorem 1 could be replaced with the following lemma.

LEMMA 3. Let l and m be partitions contained in an l Â k rectangle such that ‘ðlÞþ‘ðmÞ 4 l. Then sl Á sm ¼ðOl Á OmÞ 1.

Proof. If d 5 1 and n is a partition such that jljþjmjþjnj¼lk þ nd, then any

intersection Ol^ ðFÞ\Om^ ðGÞ\On^ ðHÞ of general Schubert varieties in Grðl þ d; E Þ must beemptysince jl^jþjm^jþjn^j 5 jljþjmjþjnjÀ2dl ¼ lk þ dk À dl > ðl þ d Þ

ðk À d Þ. This shows that hOl; Om; Onid ¼ 0. &

A dual version of this lemma states that if l1 þ m1 4 k, then sl Á sm ¼ðOl Á OmÞ 1. This follows by replacing the span of a curve with its kernel in the above proof, or by using theduality isomorphism QHÃGrðl; E ÞﬃQHÃGrðk; E ÃÞ.

6. The Minimal Power of q in a Quantum Product As a further demonstration of the use of the span of a rational curve in X, wewill give a short proof of a recent theorem of Fulton and Woodward [10] in the Ã case of Grassmannians. It generalizes the fact that any product sl Á sm in QH X is nonzero. ^ Given a partition l and an integer d 5 0, welet l denote the partition obtained by removing the top d rows and theleftmost d columns from l. Recall that a product Ã Ol Á Om is non-zero in H Grðl; E Þ if and only if li þ mj 4 k for i þ j ¼ l þ 1.

THEOREM 3 (Fulton and Woodward). The smallest power of q which appears in Ã Ã a product sl Á sm in QH X is equal to the smallest d for which O^ Á Om 6¼ 0 in H X. l d Proof. If sl Á sm contains q times a Schubert class then some curve of degree d meets each of the Schubert varieties OlðFÞ and OmðGÞ where F and G are general ﬂags. If W E has dimension l þ d and contains thespan of this curvethen W lies in

the intersection Ol^ ðFÞ\Om^ ðGÞ in Grðl þ d; E Þ. In particular this intersection is not Ã empty which implies that Ol^ Á Om^ 6¼ 0inH Grðl þ d; E Þ. Since this is equivalent to Ã O^ Á Om 6¼ 0inH X, this proves the inequality ‘5’ of the theorem. (If d > k then it is l also truethat O Á Om 6¼ 0.) l^ Now let d be the smallest number for which O^ Á Om 6¼ 0. Noticethat this implies l that l contains a d Â d rectangle, i.e. ld 5 d.Seta ¼ðk þ d À ld; ...; k þ d À l1Þ and let b bethepartition givenby bi ¼ maxðd À llþ1Ài; 0Þ for 1 4 i 4 l. If theYoung diagram for l is put in the upper-left corner of an l by k þ d rectangle, then a is thecom- plement of l in thetop d rows of this rectangle, turned 180 degrees, and b is the complement of l in theleftmost d columns, also turned. It follows from theLittlewood–Richardson rulethat theproduct sl Á sb contains

theclass s l ^ ¼ s l Á s^ and that s^ Á sa contains s ^ ¼ s d Á s^ . Sincethestruc- ðd Þþl ðd Þ l l ðkd Þ;l ðk Þ l Ã tureconstants of QH X are all nonnegative, this implies that sl Á sb Á sa contains the d product s l Á s d Á s^ ¼ q s^ . Since O^ Á Om 6¼ 0 by assumption, weconcludethat ðd Þ ðk Þ l l l d sl Á sm Á sa Á sb contains q times some Schubert class. In particular, at least one term of theproduct sl Á sm must involvea powerof q which is less than or equal to d. This proves the other inequality ‘4’ in the theorem. Notice that the identities we have used in this argument follow easily from Theorem 1 and the dual version of Lemma 3, combined with the classical Pieri rule. &

7. The Rim-Hook Algorithm Wewill nextrecallhow to carry out computations in thequantum cohomology ring Ã of X.Letci 2 QH X denote the ith Chern class of the dual of the tautological sub- bundle S E OX. This means that ci ¼ sð1iÞ for 0 4 i 4 l and ci ¼ 0 for i < 0 and Ã i > l. For p 5 1 we then define sp ¼ detðc1þjÀiÞ14i; j4p in QH X. Noticethat for p < n wehave sp ¼ Op 1, so this definition is compatible with our previous definition of sp. The definition implies that for every m 5 1 wehave

Xl i ðÀ1Þ smÀi sð1iÞ ¼ 0: ð5Þ i¼0

lÀ1 LEMMA 4. For any p 5 k þ 1 we have sp ¼ðÀ1Þ q spÀn. Proof. Both sides of the identity are zero for k þ 1 4 p < n. It follows from l lÀ1 lÀ1 (5) with m ¼ n that sn þðÀ1Þ sk sð1lÞ ¼ 0, so sn ¼ðÀ1Þ sk sð1lÞ ¼ðÀ1Þ q by Theorem 1. For p > n weﬁnally obtain Xl Xl iÀ1 lÀ1 iÀ1 lÀ1 sp ¼ ðÀ1Þ spÀi sð1iÞ ¼ðÀ1Þ q ðÀ1Þ spÀnÀi sð1iÞ ¼ðÀ1Þ q spÀn i¼1 i¼1 by induction, which proves the lemma. &

If I ¼ðI1; I2; ...; IpÞ is any sequence of integers we set sI ¼ detðsIiþjÀiÞ14i; j4p in Ã QH X. This is compatiblewith our notation sl for partitions l by Theorem 2. Any element sI can be rewritten as zero or plus or minus sm for somepartition m by moves of the form sI;a;aþ1;J ¼ 0 and sI;a;b;J ¼ÀsI;bÀ1;aþ1;J, which correspond to interchanging rows in the determinant deﬁning sI. Noticethat if l is a partition then 0 wehavetheformal identity s ¼ detðc 0 Þ where l is theconjugatepartition of l, l liþjÀi obtained by mirroring the Young diagram for l in its diagonal. In particular, if ‘ðlÞ > l then the top row of the latter determinant is zero, and sl ¼ 0.

COROLLARY 1. Let l be a partition with at most l parts. For each 1 4 j 4 l, choose Ij 2 Z such that Ij lj ðmod nÞ and j À l 4 Ij < j À l þ n. Then we have

dðlÀ1Þ d Ã sl ¼ðÀ1Þ q sI 2 QH X P where I ¼ðI1; ...; IlÞ and nd ¼jljÀ Ij.

Notice that the moves described above will rewrite the chosen element sI to zero Ã or plus or minus a basis element sm of QH X with m contained in an l Â k rectangle. The corollary is equivalent to the dual rim-hook algorithm of [2]. The formulation in terms of reduction modulo n is dueto F. Sottile[19], who in turn was inspiredby the work of Ravi, Rosenthal, and Wang [16]. This formulation gives an efﬁcient algorithm to determine if two elements sl and sm areidentical. In particular it shows that sl ¼ 0 if and only if li À i lj À j ðmod nÞ for some i 6¼ j.

Corollary 1 makes it easy to do computations in QHÃX (cf. [2, 11, 20] and [12, Ex. 13.35]). In order to expand a productP sl Á sm, one uses the classical Littlewood– n Richardson ruleto write sl Á sm ¼ clmsn where the sum is over partitions n of n length at most l and clm is the Littlewood–Richardson coefﬁcient. Each of the ele- Ã ments sn can then be reduced to a basis element of QH X by thecorollary.

8. Generators and Relations Finally, we show how to deduce Siebert and Tian’s presentation of the quantum cohomology ring of X [18] (see also [21]). We claim that

Ã l QH X ¼ Z½c1; ...; cl; q=ðskþ1; ...; snÀ1; sn þðÀ1Þ qÞ: ð6Þ In fact, all of the given relations vanish in QHÃX by Lemma 4. On the other hand, the proof of Lemma 4 together with Corollary 1 shows that the linear map Ã l d H X Z½q!Z½c1; ...; cl; q=ðskþ1; ...; snÀ1; sn þðÀ1Þ qÞ which sends Ol q to d q sl is surjective, which proves the isomorphism. Siebert and Tian gave this presen- Ã k tation in its dual form QH X ¼ Z½s1; ...; sk; q=ðY~lþ1; ...; Y~nÀ1; Y~n þðÀ1Þ qÞ where ~ ~ ~ ~ Pthe Yi are deﬁned inductively by Y0 ¼ 1, Yi ¼ 0 for i < 0, and Yi ¼ k jÀ1 ~ j¼1ðÀ1Þ sjYiÀj for i > 0. Ã ðl;nÞ The presentation (6) can also be stated as QH X ¼ Z½c1; ...; cl=I where ðl;nÞ I ¼ðskþ1; ...; snÀ1Þ. In this form thevariable q is represented by lÀ1 ðl;nÞ ðÀ1Þ sn ¼ sðkþ1;1lÀ1Þ. In [11] other generators for the ideal I are used, namely all elements sl for which l1 À ll ¼ k þ 1. It is an easy exercise to show that these d elements in fact generate the same ideal, and to show that the basis fq sl j l1 4 kg Ã of QH X is equal to the set fsl j l1 À ll 4 kg which is the basis used in [11]. We refer to [19] for a more detailed discussion of the relations of quantum cohomology of Grassmannians with other ﬁelds.

Acknowledgements We thank F. Sottile for triggering our search for simpliﬁcations to the theory of quantum cohomology for Grassmannians, when he explained his simpliﬁed rim- hook algorithm to us. We also thank W. Fulton for numerous helpful comments to our paper, some of which greatly enhanced the clarity of our proof of the quantum Giambelli formula. Finally we thank A. Kresch for simplifying our proof of the key Lemma 1.

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