Equivariant Quantum Cohomology of Homogeneous Spaces

Table of contents :
DEDICATION ............................................................................................................................ ii
ACKNOW LEDGEMENTS .................................................................................................... iii
LIST OF FIGURES ................................................................................................................ vi
LIST OF APPENDICES ....................................................................................................... vii
CHAPTER
1. Intro d u ctio n ................................................................................................................... 1
1.1 Motivation ............................................................................................................ 1
1.2 (Imprecise) Statement of results ........................................................................ 3
1.3 Statement of results - for experts ..................................................................... 5
1.3.1 Definitions and notations for general X = G / P ................................. 5
1.3.2 An equivariant quantum Chevalley rule and an algorithm ............... 7
1.3.3 Positivity ............................................................................................. 9
1.3.4 An equivariant quantum Giambelli formula for the Grassmannian . 9
1.4 Structure of the thesis .......................................................................................... 13
2. Prelim inaries ................................................................................................................... 14
2.1 Classical cohomology - establishing notations ................................................... 14
2.1.1 Roots and lengths .................................................................................. 14
2.1.2 Cohomology .......................................................................................... 15
2.1.3 (Schubert) Curves, divisors and degrees ... ........................................... 16
2.2 Equivariant cohomology ....................................................................................... 17
2.2.1 General fa c ts ....................................................................................... 17
2.2.2 Equivariant Schubert calculus on G / P ............................................. 19
2.3 Quantum cohomology .......................................................................................... 23
2.4 Equivariant quantum cohomology of the homogeneous spaces ......................... 25
3. Equivariant quantum Schubert calculus ............................................................... 28
3.1 The equivariant quantum Chevalley rule ........................................................ 28
3.2 Two formulae ...................................................................................................... 33
3.3 An algorithm to compute the EQLR coefficients ............................................ 39
3.3.1 Remarks about the algorithm ............................................................ 42
3.4 Consequences in equivariant cohomology of G/P .............................................. 43
3.5 A brief survey of the algorithms computing the equivariant or quantum
Littlewood-Richardson coefficients ..................................................................... 45
3.6 Appendix - Proof of the Lemma 3.11 ............................................................... 48
4. Positivity in the equivariant quantum Schubert calcu lu s ................................ 52
4.1 Preliminaries ......................................................................................................... 52
4.2 Proof of the positivity Theorem ......................................................................... 55
5. Equivariant quantum cohomology of the Grassmannian ................................... 59
5.1 General facts ......................................................................................................... 59
5.1.1 Definitions and notations for partitions ............................................. 60
5.1.2 Schubert varieties ................................................................................. 61
5.1.3 Equivariant cohomology ..................................................................... 62
5.1.4 Equivariant quantum cohomology ...................................................... 63
5.2 A vanishing property of the EQLR coefficients ................................................ 64
5.3 Equivariant quantum Chevalley-Pieri rule ......................................................... 70
5.4 Relation between the two torus actions ............................................................ 73
5.5 Computation of EQLR coefficients for some “small” Grassmannians ............ 75
5.5.1 The algorithm for the Grassmannian - revisited .............................. 75
5.5.2 Computation of the coefficients for Gr(2,5) .............................. 76
5.5.3 The coefficients for small Grassmannians ................................. 77
5.5.4 Multiplication table for QH^.(Gr(2,4 )) ............................................. 77
6. Polynomial representatives for the equivariant quantum Schubert classes
of the Grassm annian ........................................................................................................ 79
6.1 Factorial Schur functions .................................................................................... 79
6.2 Proof of the formulae .......................................................................................... 84
6.2.1 A characterization of the equivariant quantum cohomology ........... 85
6.2.2 An equivariant quantum Giambelli and presentation ..................... 86
A PPEN DICES ................................................................................................................................. 94
BIBLIOGRAPHY ........................................................................................................................... 116

Citation preview

Equivariant quantum coh om ology o f h om ogeneous spaces

by Constantin Leonardo Mihalcea

A dissertation subm itted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2005

Doctoral Committee: Professor Professor Professor Assistant

William E. Fulton, Chair Sergey Fomin Robert K. Lazarsfeld Professor Jamie Tappenden

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

UMI Number: 3186707

INFORMATION TO USERS

The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.

®

UMI UMI Microform 3186707 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

To m y parents (Parin^ilor m ei)

ii

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

ACKNO W LED GEM ENTS

Mathematically, I have benefitted from conversations with many people. Among them I would like to mention Arend Bayer, Gilberto Bini, David Bortz, Calin Chindri§, Gabi Farca§, Milena Hering, Paul Horja, Bogdan Ion, Alex Kuronya, Nguyen Minh, Evangelos Moroukos, Mircea Mustafa, Tom Nevins, Sam Payne, Mike Roth, Janis Stipins and Alex Yong. Special thanks are due to Sergey Fomin, Allen Knutson and Ravi Vakil who helped me become a better mathematician. Most of all I am indebted to my advisor, Professor William Fulton, who guided and inspired me throughout the graduate school. Personally, I would like to thank my parents and my parents in law, and especially my wife, Stanca Ciupe, for being the way she is.

iii

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

TABLE OF CO NTENTS

D E D IC A T IO N ............................................................................................................................

ii

A C K N O W L E D G E M E N T S ....................................................................................................

iii

LIST OF F IG U R E S

vi

................................................................................................................

LIST OF A P P E N D IC E S

.......................................................................................................

vii

CHAPTER 1. I n t r o d u c t i o n ................................................................................................................... 1.1 1.2 1.3

1

M otivation............................................................................................................ (Imprecise) Statement of r e s u lts ........................................................................ Statement of results - for experts ..................................................................... 1.3.1 Definitions and notations for general X =G / P ................................. 1.3.2 An equivariant quantum Chevalley ruleandan a lg o rith m ............... 1.3.3 Positivity ............................................................................................. 1.3.4 An equivariant quantum Giambelli formula for the Grassmannian . Structure of the thesis..........................................................................................

1 3 5 5 7 9 9 13

2. P re lim in a rie s ...................................................................................................................

14

1.4

2.1

Classical cohomology - establishing notations................................................... 2.1.1 Roots and lengths.................................................................................. 2.1.2 Cohomology.......................................................................................... 2.1.3 (Schubert) Curves, divisors and degrees.............................................. Equivariant cohomology....................................................................................... 2.2.1 General f a c t s ....................................................................................... 2.2.2 Equivariant Schubert calculus on G / P ............................................. Quantum cohomology.......................................................................................... Equivariant quantum cohomology of the homogeneousspaces.........................

14 14 15 16 17 17 19 23 25

3. E quivariant q u an tu m S chubert c a lc u lu s ...............................................................

28

2.2

2.3 2.4

3.1 3.2 3.3

The equivariant quantum Chevalley rule ........................................................ Two formulae ...................................................................................................... An algorithm to compute the EQLR coefficients............................................ 3.3.1 Remarks about the alg o rith m ............................................................ 3.4 Consequences in equivariant cohomology of G / P .............................................. 3.5 A brief survey of the algorithms computing the equivariant or quantum Littlewood-Richardson coefficients..................................................................... 3.6 Appendix - Proof of the Lemma 3.11 ...............................................................

iv

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

28 33 39 42 43 45 48

4. P o sitiv ity in th e equivariant quantum Schubert c a l c u l u s ................................

52

4.1 Prelim inaries......................................................................................................... 4.2 Proof of the positivity T h e o re m .........................................................................

52 55

5. Equivariant quantum cohom ology o f th e G rassm an n ian ...................................

59

5.1

General fa c ts ......................................................................................................... 5.1.1 Definitions and notations for partitions............................................. 5.1.2 Schubert varieties................................................................................. 5.1.3 Equivariant cohom ology..................................................................... 5.1.4 Equivariant quantum cohomology...................................................... A vanishing property of the EQLR coefficients................................................ Equivariant quantum Chevalley-Pieri ru le ......................................................... Relation between the two torus a c tio n s ............................................................ Computation of EQLR coefficients for some “small” Grassmannians............ 5.5.1 The algorithm for the Grassmannian - revisited.............................. 5.5.2 Computation of the coefficients for Gr(2 ,5 ).............................. 5.5.3 The coefficients for small G rassm annians................................. 5.5.4 Multiplication table for QH^.(Gr(2,4 ) ) .............................................

59 60 61 62 63 64 70 73 75 75 76 77 77

6. P olyn om ial representatives for th e equivariant quantum Schubert classes o f th e G r a s s m a n n ia n ........................................................................................................

79

5.2 5.3 5.4 5.5

6.1 6.2

Factorial Schur fu n ctio n s.................................................................................... Proof of the form ulae.......................................................................................... 6.2.1 A characterization of the equivariant quantum cohomology........... 6.2.2 An equivariant quantum Giambelli and p re sen tatio n .....................

79 84 85 86

A P P E N D I C E S .................................................................................................................................

94

B IB L IO G R A P H Y ...........................................................................................................................

116

v

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

L IS T O F F IG U R E S

Figure

5.1

Example of a partition: p = 3, m = 7, A = ( 4 ,2 ,1 ).......................................................

60

5.2

Example of partitions A and A- ......................................................................................

61

5.3

The set of vertical steps of a partition: p = 3, m = 7 ,7(A) = { 1 ,4 ,6 } ......................

71

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

L IS T O F A P P E N D IC E S

A p p end ix

A.

Gysin maps ..........................................................................................................................

95

A.l The non-equivariant c a s e .................................................................................... 95 A.2 Finite dimensional approximations and equivariantGysin maps ......................102 B.

Equivariant quantum cohomology - generaldefinition and proofs of itsproperties . . 106

vii

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

CHAPTER 1

In tr o d u c tio n

1.1

M otivation

Inspired by physicists, a new deformation of the cohomology of a (smooth) variety X , called (small) quantum cohomology, has been recently constructed. It encodes certain enumerative information about the variety X . Understanding quantum co­ homology in general proved to be a very difficult task and it is known only in some limited cases (see e.g. [1 ] and references therein). It has been better understood when X is a homogeneous space G / P , where G is a complex, semisimple, connected Lie group and P a parabolic subgroup of G. In this case the classical cohomology of X is an algebra with an additive basis consist­ ing of Schubert classes, and the multiplicative structure constants are the famous (generalized) Littlewood-Richardson coefficients, which play a fundamental role in various ares of mathematics, such as algebraic geometry, algebraic combinatorics, representation theory. The quantum cohomology of A = G / P is an algebra which is a deformation of the classical cohomology. It has an additive basis determined by the Schubert classes of X , and its multiplicative structure constants, which generalize the LittlewoodRichardson coefficients, are the (3-point, genus 0) Gromov-Witten invariants. These

1

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

2

are positive integers equal to the number of rational curves of a certain degree pass­ ing through three Schubert varieties in general position, whose dimensions add up to the dimension of X . The associativity of quantum cohomology gives a systematic approach for finding such numbers. In fact, since X is a homogeneous space, the situation is more fortunate. The richness of the geometry and of combinatorial prop­ erties of X = G / P have been used to obtain better algorithms for the computation of the Gromov-Witten invariants, which are particularly efficient when G is of type A (e.g. G = PGL{m)){5, 25, 19, 16, 28, 57, 73], However, even in this case, the story is far from complete, as there is no proven positive combinatorial formula for these coefficients. In fact, in spite of much recent progress [6 , 6 8 , 13, 16, 21] the only case in which there is such a (conjectural) formula is the type A Grassmannian ([16]). The initial motivation for the project undertaken in this thesis was to understand better the quantum cohomology of homogeneous spaces in general, and, in particular, to find a positive formula for the Gromov-Witten invariants of the Grassmannian. In order to do that, we have considered an equivariant version of quantum cohomology, named equivariant quantum cohomology. It was introduced by A. Givental and B. Kim [32] with the same purpose - to study the quantum cohomology of homogeneous spaces. It turned out however th a t the equivariant quantum cohomology has many inter­ esting properties in its own right, therefore deserving a closer study. Besides this, one also gains a better insight into the (non-equivariant) quantum cohomology. In this respect, we obtain a new algorithm to compute the Gromov-Witten invariants. Its main ingredient is a certain recurrence formula satisfied by the structure constants of the equivariant quantum cohomology ring. This formula seems to play a key part

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

in the study of the conjectural positive formula for the Gromov-Witten invariants. A first, imprecise, version of the results is given in the next section, with references to the precise statements. Another section, intended for the experts in the field, which contains precise statements, is also part of the introduction. All of these results have been announced by the author in [59, 60, 61, 62]. 1.2

(Im precise) S tatem en t o f resu lts

Let X be the homogeneous space G /P , where G is a connected, complex, semisim­ ple Lie group and P a parabolic subgroup of G. The goal is to study an equivariant version of the (small) quantum cohomology of the homogeneous space X , called equivariant quantum (EQ) cohomology. Fix P D B D T, a parabolic subgroup P containing a Borel subgroup B, which in turn contains T , its maximal torus. The (small) T —equivariant quantum cohomology of the homogeneous space X is a deformation of both equivariant and quantum cohomology rings. It is a graded algebra over A[5 ], where A is a polynomial ring (the T —equivariant cohomology of a point), and q stands for a sequence of indeterminates indexed by the simple roots with respect to B whose corresponding reflections are not contained in the Weyl group of P. The equivariant quantum cohomology has a A[g]—basis consisting of Schubert classes, and the multiplication is determined by the equivariant Gromov-Witten invariants. The structure constants of equivariant and quantum cohomology form a certain subset of these invariants. The invariants from the complement of this subset will be called m ixed. The first result of this thesis is an EQ Chevalley rule in the EQ cohomology of X , i.e. a rule for multiplying by a divisor Schubert class: T heorem 1.1. The Chevalley rule for the equivariant quantum multiplication in X

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

4

does not contain any mixed terms. The equivariant and the quantum term s th a t appear in this rule are known (see [48,

8,

3] for the equivariant case and [28] for the quantum case), so Theorem 1.1

implies a precise formula (see Thm. 3.8 below). It turns out th a t this multiplication by divisors is enough to determine the EQ cohomology algebra, even though, unless P = B, the divisor classes do not generate it. T heorem 1.2. Let A be a free graded A[q}—module with a distinguished basis { m — p and e* = 0 if i
p.

By Hk for m —p + 1 ^ k ^ m (respectively Eh, for p + 1 ^ k ^ m), we denote the determinants th a t express the factorial homogeneous Schur functions (respectively, factorial elementary Schur functions) via the Jacobi-Trudi formula: Hh

d e t(rJ ®i+j—i)i^»j^fc,

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

12

E k — d e t(r

fi1+j —i) 1 ^ j r-

For A a partition in the p x (m —p) rectangle define the factorial Schur determinants (cf. property (D), equation 6.1.7 below):

sA = d e t(r 1- -7/iA4+j-i)i Wr (X ) for any integer i. In certain situations, the map / induces also a Gysin map f j : WT (X ) — ♦ H fl 2d(Y) where d = dim A — dim Y . This will happen if Y is smooth or if / is a regular

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

19

embedding. The definition and some of the properties of this Gysin map are discussed in Appendix A. N o ta tio n 2.1. If Y is a point then this Gysin map is denoted by f x . 2.2.2

Equivariant Schubert calculus on G / P

In this section X denotes the homogeneous space G /P . Note th a t the Schubert va­ rieties X (w ) and Y (w ) are T —invariant. Since X is smooth, these varieties determine equivariant cohomology classes a(w)T = [AT(u;)]r in H ^ w\ x ) and a(w)T = [Y(w)}r in H ^ W\ X ) . Contrary to the non-equivariant case, a(wv )T is not equal to a{w)T. In fact, there is an isomorphism Tp : H f ( X ) —> H f ( X ) sending [X ('u;v)]r to [Y(w)}T, induced by the involution tp : X —>• X given by tp(:v) = w 0 ■x. This map is not T —equivariant, but it is equivariant with respect to the map T —> T defined by t —> w0twQ1 = w 0tw0, hence over

the isomorphism Ip sends c(x) to c(wox),

where (w 0x ) ( t ) = x(m0tm0). Since {&(w)}weWp (resp. {5(m)}UJ6M/p) is a basis for the classical cohomology of X , the Leray-Hirsch Theorem ([39] C h.16), applied to the A —bundle X T —> B T , implies th a t the set {&{w)T}weWp (resp. T —equivariant cohomology of X .

{&{w)T}weWp) is a basis for the

There is an equivariant version of the duality

theorem, described using the H ^(pt)—bilinear pairing (-, -)T : H*t {X) ®H,Tipt) H * (X ) — ♦ H*T (pt) given by x

0

y —> Jx x U y . One has the following result:

P ro p o s itio n 2.2 (Equivariant Poincare Duality). The bases { B T , discussed in the Appendix (§A.2). We just recall th a t X T>n denotes the quotient (X x E T n) / T and th a t the Schubert varieties X ( u ) and Y (v) determine varieties X (u)t,u respectively Y(v)x.n, hence cohomology classes

a

( u ) t,«

€

H 2c(-u\ X T^n) respectively cf(v)T,n € H 21^ (XT,n) ■

For n large enough, HT l (X ) = I P ( X T>n) and the equivariant Gysin map is defined via the ordinary one, using the spaces X TtnThis definition of the equivariant Gysin map shows th at it is enough to prove the result for the finite dimensional approximation versions of the classes involved. It is a general fact th a t the restrictions of X { u)t ,u (respectively Y {v)T,n) to a fiber 71%^(6) of the map 7 r : X T)1l —» BT n, where b belongs to B T n, is isomorphic to the ordinary Schubert variety X ( u ) (resp. Y(v)) in 'n'/l/b) — X . The Schubert varieties X (u ) and Y (v) intersect properly (this is a consequence of the Kleiman’s transversality Theorem, see e.g. [28] Lemma 7.2, recalled in lemma 3.3 below). If this intersection is positive dimensional, the image through 7xr,n of X (u ) C\Y(v) has dimension strictly less than the source, which shows th a t (-/rr,n)*(o'(^)r,n U &(v)T,n) = 0. It remains to consider the case when the intersection of X (u ) and Y (v) is 0—dimensional. Then the intersection X (u ) fl Y (v) is empty (thus (7TT,n)*(&(u)T,n U 3f(w)r,n) == 0 again) unless u = v when X (u ) and Y (v) intersect transversally in a single T —fixed

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

21

point uP. Using th at, one gets a T —invariant section s : B T n — ►X T .n sending b in BTn to u P in 7T—1(&). Then

(irT,n)*(cr(u)T,nVv(v)T,n) =

1)) = 1

where (the first) 1 is the cohomology class determined by the fundamental class of

□ As in the non-equivariant case, this duality implies a formula for the equivariant Littlewood-Richardson (ELR) coefficients c'/v obtained from the expansion

W

They can be computed as follows: (2 .2 .1)

From either of these descriptions it follows th a t the ELR coefficient c™v is a polyno­ mial in H^(pt) = Z[xlt ...,xr\ of degree c(u) + c(v) —c(w). Proving a conjecture of D. Peterson ([67]), Graham showed th a t the ELR coefficients c™v are polynomials in the negative simple roots x lt ...,xr with nonnegative coefficients (cf. [35]). Remark 2.3. G raham ’s result deals with the case X = G /B . The more general situ­ ation X = G / P follows from the fact th a t the T —invariant projection p : G / B — > G / P induces an injective map p*T : H ^{G /P ) — >H*r {G/B) in equivariant cohomol­ ogyA positive combinatorial formula for these coefficients was obtained in [46] when X is a Grassmannian. The key to th a t was a certain recursive formula for the ELR coefficients, which holds in th a t case (cf. [64, 65, 46]). Another recursive formula for any G / B was obtained in [44],

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

22

We recall next the formula for the special multiplication with a divisor class, which will be generalized later. D efin itio n 2.4 (Definition of the equivariant coefficient c f ^ w). Let w be a min­ imal length representative in W p , and let w = .ssti ■... • spik be a reduced word decomposition. For $ a simple root in A \ A P, define the linear form D(spi,w) to be: D (s 0 i ,w) = 'y spu • ... ■ ij=i

(2.2.2)

It can be shown th a t each term of the sum is a positive root a in W with the prop­ erty th a t w~xa is a negative root (see the Appendix to Chapter 3 for the positivity statement and e.g. [37] §1.7 for the second one). Let ip : A — > A be the automor­ phism sending the positive simple root ft to the negative simple root 'ii’o(P) (see the paragraph before Cor. 3.27 for a proof of that). Then the equivariant coefficient c7(p),w is eciual t0: c:(0hw = (p(D{s0, wv)).

(2-2.3)

We are ready to state the equivariant Chevalley formula (cf. [50] Thm. 11.1.7(c) and Prop. 11.1.11): P ro p o s itio n 2.5 (Equivariant Chevalley formula). The following formula holds in H*t {X): (2.2.4)

o-(s(/?))T • a(w)T = ^

ha(u 0 )a(wsa)T + cw s(0 lwa{w)T

where the sum is over all a € P such that wsa is a representative in W p of codimension c{wsa ) = c{w) + 1.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

23

2.3

Q uantum cohom ology

The goal of this section is to recall the basic definitions associated to the small quantum cohomology of a homogeneous space G /P . A more general approach, for any X smooth and convex, is given in Appendix B in the context of equivariant quantum cohomology. The (small) quantum cohomology of X — G / P is a graded Z[q]—algebra, having a Z[q\—basis consisting of Schubert classes a(w), for w in W p . Here q stands for the indeterminates sequence (qp), indexed by a basis of H 2 (X ), hence (recall) by the simple roots /? in A \ A P. The complex degree of qp is (2.3.1)

deg qp — [

Ci(TX)

JX {s0)

where T X is the tangent bundle of X (for an explicit computation of this degree, see e.g. [28] §3). For a general degree d = ^2dpa(sp), qd stands for the monomial \ \ q dp and deg qd denotes ]T] npdp. As a convention, all the degrees will be complex. If a is a positive root not in )+]C deg qp-dp. We call the coefficient

a quantum Littlewood-Richardson coefficient.

The fact th a t such a multiplication gives an associative operation was proved, using algebro-geometric methods, in [47] (also see [71] and references therein). We will also give a proof in Appendix B, in the more general context of equivariant quan­ tum cohomology. Computations and properties of the (small) quantum cohomology algebra were done e.g. in [72, 5, 6 , 6 8 ] for Grassmanians, [20, 17, 25, 28, 73] for (par­ tial) flag manifolds and in [49] for the Lagrangian and orthogonal Grassmannian. Recently, a simplification of the methods used to prove results about the quantum cohomology of the Grassmannian was achieved by Buch ([13]). Similar ideas were since used in [15, 14, 16] for more general situations. We recall an equivalent definition of the coefficient c ^ , which will be generalized in the next section. Let M o }3 (X ,d) be Kontsevich’ moduli space of stable maps. This is a projective algebraic variety of dimension dim A + ^2 dp ■(deg qp), whose (closed) points consist of rational maps / : (C ,px,p 2 ,p3) — > X of multidegree d, where C is a tree of IP1 ’s.

There are evaluation maps evi : Ado,3 (X ,d) — > X

which send a point (C,px,p 2 ,p3; f ) to f(pi) (i = 1,2,3), and a contraction map 7T : A4 o,3 (X ,d) — > Ado,3 — pt (for details see e.g. [71]). Then the quantum LR coefficient is given by (2.3.3)

c^=

[_

evla(u)Gev* 2 a (v)G evla (w )

JM oAX,d)

in H°(pt). We also recall the quantum Chevalley rule proved in [28], Thm. 10.1. P rop osition 2.6 (quantum Chevalley). Let P be a simple root in A \ A p and w a

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

25

minimal length representative in W p . Then (2.3.4)

a(s(/3)) * a{w) =

ha{up)■ Y be a T —equivariant morphism of two algebraic

varieties, with Y smooth. Let V be a T —invariant subvariety o f Y of codimension c and let \V]t € H fff Y ) be the equivariant cohomology class o f V . Then the equivariant cohomology pull-back F^([V] t ) is equal to 0 if F ~ l {V) is empty. Proof. From the definition of the equivariant cohomology using the finite-dimensional approximations, it is enough to show th a t enough. Note th a t

F ~ l (V)

F f n ( [ Vr , n])

is empty implies th a t

= 0 in

H 2c ( Y r tn),

F ~ l ( V r JL)

for n large

is empty.

28

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

Let e =

29

dim Y —dim V. Recall th a t the cohomology class [Vr,n] G H 2c(YT .n) is, by definition, the restriction of the fundamental class [V] € H 2e( Y ,Y \ V) to H 2e(Y). Then the lemma follows from the commutativity of the following diagram: H 2* { X ^ X > T^ F ^ n(yT,n)) ^

H 2e( X ^ n)

H 2*(YTtn,Yr,n ^ V T,n)

H 2*(YT,n)

□ The next lemma is a first step towards a vanishing property of some of the EQLR coefficients. L em m a 3.2. L e tu ,v ,w be three representatives in W p such that one of the intersec­ tions e v f 1 (X(u)) fl e v f 1 (Y(w)) or e v f 1 (X(v)) D e v f 1 (Y(w)) i n M .0 ^{X,d) is empty. Then the EQLR coefficient

is equal to zero.

Proof. The hypothesis implies th a t one of the products evf*(a(u)T) ■evp*(r(w)T) or ev2 *(a (v )T) ' evI*{T(w )T) in H ^ X ) vanishes. The assertion follows then from the definition of the EQLR coefficients (see formula

(2.4.1)from the previous chapter)

and from the lemma 3.1 above.

□

The following Lemma, inspired from [28], uses a weaker version of Kleiman’s transversality Theorem ([43]) to show th a t the intersection of the inverse images of two opposite Schubert varieties through a G—equivariant map has the expected dimension. This is in fact the key lemma used in the proof of the vanishing of the mixed EQLR coefficients from the EQ Chevalley formula (see Lemma 3.5 below). L em m a 3.3. Let Z be a reduced, possibly reducible, pure dimensional G-variety and let F : Z — > X x X be a G —equivariant morphism, where G acts diagonally on

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

30

X x X . Then, for any u and v in W p , the subscheme F ~ 1 ( X ( u ) x K(u)) is either empty or of codimension c(u) + I (v). Proof. Note th a t every irreducible component of Z must be G —invariant. Indeed, the action of G permutes the irreducible components, and the unit element in G fixes each component. This implies th a t it is enough to show the result in the case when Z is irreducible, which is a part of what is proved in Lemma 7.2 from [28]. For convenience, we summarize its proof. Kleiman’s transversality result ([43], Thm. 2) yields an open subset U in G x G, invariant under the diagonal left-multiplication by G, such th a t F ^ ( h i X ( u ) x h 2X ( v v )) is either empty or of codimension c(u) + l(v) for any ( h \,h 2) in U. Another lemma ([28], Lemma 7.1), shows th a t for any pair (5 1 , 9 2 ) in G x G such th a t the intersection g i B g f 1 fl g2 B g f 1 is a maximal torus in G, there is a pair (hi, ha) in U such th at h \X (u ) = g \X (u ) and h 2X ( v y ) = g2X ( v y ). The result follows then by taking g\ = 1 and g2 = w0.

□

Remark 3.4. Since only the dimension assertion of the Kleiman’s Theorem is used, the lemma is valid in all characteristics. However, it will be used for Z being the moduli space of stable maps A4 0i3 (X, d), whose construction is done in characteristic zero. We are ready to prove the main vanishing result for the EQLR coefficients. Recall th a t np denotes the complex degree of the indeterminate qp, for (3 in A \ A P. L em m a 3.5 (Main Lemma). Let u , v ,w be representatives in W p and d = (dp) a nonzero degree such that c(u) + 1 > c(w) + ^Znpdp. Then the EQLR coefficient cffif is equal to zero.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

31

Proof. By Lemma 3.2 it is enough to show th a t the intersection E(u, v ) := e v f 1( X (tt))fl e v f 1 (Y(w)) in

d) is empty. The hypothesis implies th a t c(u) + I (w) + 1 > dim X + ^

npdp = dim Ado,3 (X , d)

which is equivalent to c(u) + l(w) ^ dim Ado,3 (X, d). Lemma 3.3 applied to Z — M 0 fl(X,d) and F

F

: M o !3 (X ,d) — *• X x X given by

— (evi,evz) implies th a t the intersection in question is at most finite. Moreover,

the boundary B of Ad0,3 (X ,d), which is the subvariety consisting of stable maps (C,Pi,P 2 ,P3 ',f) where the curve C has at least one node is G —invariant and it is of codimension one. Applying again Lemma 3.3 for Z = B and

F

restricted to B,

shows th a t if E(u, v ) is not empty then it cannot intersect B, so all its points must be stable maps whose sources are curves isomorphic to P 1. One the other side, given a stable map / : (C ,p i,p 2 ,Ps) — ►X in Ad0,s(X, d) such th a t f(p i) is in X ( u ) and f(ps) is in Y (w ) and C ~ P 1, one can produce a curve in E(u, v ) by letting / ( p 2) to vary in the image of C through / (this image is not a point, since the degree d is not zero). This constitutes a contradiction with the fact th a t E ( u , v ) is finite.

□

Remark 3.6. W hen X is a Grassmannian, we will prove a stronger vanishing property in §5.2. Remark 3.7. It is known th a t the moduli space Ado,3 (X ,d) is irreducible ([42, 70]), so we could have used to original version of Lemma 3.3, where Z was assumed irreducible. An immediate consequence of the Main Lemma is th a t all the mixed EQLR coef­ ficients in the product a(s(f3))oa(w) must vanish. Indeed, by definition, a coefficient

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

32

cs(t) u

m'xecl ^ the degree d = (dp) is not zero and if its polynomial degree is also

not zero, i.e. if c(u) + 1 > c(w) + Main Lemma, so

ripdp. This is precisely the hypothesis of the

u = 0. In particular, combining the quantum and equivariant

Chevalley formulae (2.3.4) and (2.2.4) yields Theorem 3.8 (Equivariant quantum Chevalley rule). Let (3 be a simple root in A \ A p and w a minimal length representative in W p . Then the following formula holds in the equivariant quantum cohomology of X = G /P : (3.1.1) a(s(/3)) o a(w) = E

ha(u>p)a(wsa) + E

qd{a)ha(ujp)a(wsa) + cf{/3) wcr(w)

where the first sum is over all positive roots a in $ + \ $ p such th atw sa is a represen­ tative in W p and c(wsa) = c(w) + 1 , and the second sum is over those a in 4>+ \ such that w sa is a representative in W p of codimension c(wsa) = c(w) + 1 —n(a). The coefficient c f ^ w is the one given in Prop. 2:5. The equivariant quantum Chevalley formula implies a recursive formula satisfied by the EQLR coefficients. Using double recursion, first on the degree d, then on the polynomial degree, this formula shows th a t the EQLR coefficient c™'d is equal to a homogeneous combination of EQLR coefficients, some with smaller degree d, and the remaining ones with the same degree d but higher polynomial degree. Corollary 3.9. Let u , v ,w be representatives in W p , /? a positive simple root in A \ Ap and d a degree. Then the following formula holds:

a

E cx,d^d(a)

a.

h^ K

i 1v d{a)-

E

M ^ K 2/ ~ d(a)

a,d ^d((x )

where the first two sums are over a € 4>+ \ p such that ?q = u.sa is a representative in W p of codimension c(ui) = c(u) + 1 and wi is such that w \sa = w and c(w) =

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

33

c(wi) + 1; the last two sums are over a € (f>+ \

with d ^ d(a) (with d(a) defined

by (2.1.1)) such that U2 = usa is a representative in W p with c(u\) = c(u) + 1 —n ( a ) and W2 is a representative such that w = w 2 sa and c(w) — c(w2) + 1 —n(a). Proof. This is a straightforward computation and it follows from collecting the coef­ ficient of qda(w) in both sides of the associativity equation (a(s((3)) o a(u)) o a(v ) = a(s(/3)) o (cr(ti) o a(v)). It will be done in the (notationally) simpler case when X is a Grassmannian (see Prop. 5.12 below).

□

Remark 3.10. This formula is the main ingredient for an effective algorithm to com­ pute the EQLR coefficients, found in §3.3 below. 3.2

Tw o form ulae

The aim of this section is to prove two formulae satisfied by the EQLR coefficients, which will be used in the algorithm of the next section.

From now on, all the

results will be algorithmic, and a coefficient c™’d is regarded as a (possibly rational) homogeneous function of degree c(u) + c(v) — c(w) — Y nf)dfj- The latter quantity will still be called polynomial degree, even though it may a priori be negative. To state the formulae, we need to define a reversed Bruhat ordering denoted -< on the perm utations W p as follows: write w\ —» w 2 if there exists a a positive root in + \ p such th a t w 2 — Wisa and c(w 2 ) > c(wi). Then u -< w if there is a chain u = w 0 —> Wi —» ... —> Wk = w (in fact, in the definition of W\ —►W2 , §5.11 in [37] shows th a t it is enough to consider those a for which c(w2 ) = c(w\) + 1).

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

34

We rewrite next the recursive formula from Cor. 3.9 as wd

K M

Fw,u(p)

' l 2 a , d ^ d ( a ) h a ( U f 3 ) c U2,v

( ^ ^

Z ) a ,d > d ( a ) h g f i o ^ C u f i l

F w ,u {P )

( )

F W!u ( ( 3 )

where w is different from u and Fw,u{(3) is the linear homogeneous form in the negative simple roots in W defined by

F W>U( P ) =

c 7(I3), w

~

C™(P),u

with /? a positive simple root in A \ Ap. The following Lemma gives some of the basic properties of the forms FWtU((3) to­ gether with the equivariant coefficients c f ^ w used to define it. To shorten notations, for each u, w in W p such th a t u -< w, we denote by Cov(u, w) the set of positive roots a in + \ p such th a t usa is a representative in W p , c(usa) = c(u) + 1 and usa -< w. L em m a 3.11. Let u ,w be two distinct representatives in W p . 1.

The coefficient

for (3 in A \ A p, is a linear homogeneous combination

of negative simple roots x\,

with nonnegative coefficients, and there exists

a (3 for which this coefficient is nonzero. 2. There exists a positive simple root (3 in A \ A p such that FWtU((3) is nonzero. 3. Assume that u < w in the reverse Bruhat ordering previously defined.

Then

for any (3 in A \ Ap the form FwViW(d) is algorithmically known, and it is a linear homogeneous form in EQLR coefficients of degree strictly smaller than d, with coefficients in the fraction field R(A) of A = Zfaq, ...,x m]. I f d — 0 then EUfViW(0) = 0. Proof. We argue by descending induction on c(u) — c(w) ^ dim X , with equality exactly when u is the unit element 1 in W p and w — Tu0 (the representative in W p which indexes the unit element in Q H f ( X ) ) . In this case the first two sums from (3.2.1) (applied with a suitable /?, such th a t FW(J:i(/3) doesn’t vanish, cf. Lemma 3.11) disappear and hence cf°v’d is equal to a combination of EQLR coefficients coming from the last two sums, which have coefficients of degree strictly smaller than d (thus defining £ ,lj„)™0(d)). Assume now th a t c{u) — c(w) < dim A . Since u -ft w, u cannot be equal to w, hence one can apply formula (3.2.1) to c f ’f (again, with a suitable ,8 ). The last two sums enter into the definition of E u .„,w{d). Note th a t the coefficients P f fi from the first two sums satisfy c(u) —c(v) < c(u1) —c(wi), so to finish the proof it is enough to show th a t these coefficients satisfy the induction hypothesis, i.e. th at U\ -ft Wi. Assuming the contrary, i.e. ui -< wx, since u -< iq and W\ -< w, it follows th a t u -< w, a contradiction. The case d = 0 is treated in the same way, proving now th a t c f ’f = 0 if u ft w.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

□

36

Next is a formula which shows th a t the coefficients of the form

are determined

algorithmically by coefficients of the form c ^ dw with ua -< w and by coefficients of (strictly) smaller degree d. To write this formula we will introduce a weighted Bruhattype oriented graph T(u, w, / ) , encoding all the possible saturated paths tt from u to w >- u, weighted according to the recipe described below. D efin itio n 3.13. Let u ,w be two representatives in W p such th a t u -< w. The weighted oriented graph T(u,w, f ) is given by the following data: • A set V (u ,w ), of vertices, which consists of all representatives v in W p such th a t u -< v -< w. • For each Vi, v 2 in V(u. w) such th a t v 2 = v\sa, where a is in + \ T j such th at c(v2 ) = c(v 1 ) + 1 (i.e. a is in Cov(vi,w)), there is an edge between Vi and v2, oriented from v\ to v2. This edge is denoted in short by Vi —>v2. An assignment / : V ( u ,w ) — > A \ A p, v —> (3(y) such th a t FWtV(f3(v)) is a nonzero, nonnegative combination of negative simple roots (such an / exists by Lemma 3.11, assertions (2) and (3)). For each edge v —* vsa in r(u , w; f ) define its weight to be

,/ a , ha(u}p(y)') wt(v vsa) = „

K,v(P(v))

A p a th 7Tin T(u, w; f ) is any oriented path from u to w. The weight of 7r, denoted wt{7r), is the product of all the weights of the edges it contains. To any such oriented weighted graph T ( u , w , f ) one associates a homogeneous rational function in R(w, u\ f ) in the fraction field R (A) defined by

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

37

(3.2.3)

R(u, w; f ) —

w t( tt) 7T

where the sum is over all paths in the graph T(u,u;; / ) . We will need the following Lemma: L em m a 3.14. Let u ,w two representatives in W p such that u -< w.

Then there

exists an assignment f : V ( u , w ) — ►A \ Ap as in Definition 3.13 above such that the function R(u, w\ f ) is not zero. Proof. For any assignment / , a path

in T(u, w ; / ) has weight .( \

TT

^Qi (u P(ui - 1))

i—1 Definition 3.13 and Lemma 3.11 imply th at the denominator is a nonzero, nonnega­ tive homogeneous linear combinations of negative simple roots, while the numerator is a product of nonnegative integers. Then an assignment for which R ( u , w , f ) is nonzero would be any assignment for which the graph T ( u ,w , f ) has a path tt of nonzero weight. Such an assignment is provided by (4) from Lemma 3.11. To each vertex v from V(u. w), choose a positive root a(v) in Cov(v, w). Then (3{v) is chosen to be any of the positive simple roots in A \ A p such th a t FVSalvj.,,(fi{v)) is nonzero. Indeed, in this case Fw>v(P(v)) = FWtVSa(v)((3(v)) + FV3a(v)tV(f3(v)) > 0 by Lemma 3.11 and by definition of (3(v). Given this assignment, a path tt of nonzero weight can be constructed inductively as follows: assuming th a t the (i + l ) st element ut is con­ structed, and a positive root a = a(ui) has been chosen as before, u i + 1 is equal to

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

38

UiSa. In this case, the weight of the edge u, A u l+1 is ha )) Fw,Ui{P(ui)) and ha(ujf}(ui)) is a positive integer by Lemma 3.11(4), as desired.

□

Remark 3.15. We will see (Cor. 3.20 below) th a t the function R ( u , w , f ) is inde­ pendent of the assignment / , and hence nonzero, by the Lemma just proved. This will be a consequence of the fact th a t c^’°w = 1 and of the formula (3.2.4), which is proved next. P ro p o s itio n 3.16. Assume that the EQLR coefficients are commutative, i.e. cv ’u f or any representatives u ,v ,w in W p and let

uq, w

=

be two such representatives

such that u 0 -< w. Then for all assignments f : V (u 0 ,w) — >A \ A P, the EQLR coefficient c™fw satisfies the following formula (3.2.4)

0. Applying (3.2.1) to (3.2.5)

= ( J 2 h^

and using Prop. 3.2.2 yields

0 (uo)KfSatW) / F w,uo( P ( ^ ) )

+ E'WtUO{ffiu0), d)

a

where the sum is over all a in Cov(u0, w) and E'w uo(P(uo),d) is an A(A)-linear combination of the EQLR coefficients of strictly smaller degree.

It contains the

EQLR coefficients from the last two sums of (3.2.1) since they have smaller degree,

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

39

the coefficients

fv from the second sum, since they satisfy

= E UOtWiWl(d) by

and w -ft Wi), and those coefficients c™fSa W from

Prop. 3.2.2 (because

the first sum for which a is in + \ $ p , c(u0 sa ) = c(u0) + 1 but UoSa yi, w, to which one applies again Prop. 3.2.2. Note th a t (3.2.5) is equivalent to w t(uo ^ u 0 sa) c ^ dSatW + E ^ uo{p(uo),d)

(3-2.6) a£Cov(uo,w)

Induction hypothesis, applied to each c™£aA \ A p such th a t R (w 0, w; f ) is not equal to zero (this assignment exists by Lemma. 3.14). Then cw% = l / R ( w o , w ; f ) .

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

42

Step 2: Compute all other coefficients of degree 0. We argue by descending induc­ tion on the polynomial degree. Note th a t the last two sums of (3.2.1) vanish since d = 0 and d{a) > 0. The largest polynomial degree for the coefficient dff, is achieved exactly when u = v = 1 and vj = wQ. In this case the coefficient in question is equal to 0, by Prop. 3.2.2, since 1 -fk w0. Consider now a coefficient

of smaller poly­

nomial degree. If u = v = w this coefficient is known by Step 1. By commutativity, we can assume then th a t u

w. Applying formula (3.2.1) to cffif (with a suitable fi)

writes this coefficient as an R (A )—linear combination of coefficients of polynomial degree larger by one, hence known by induction. This finishes the proof of the case d — 0. Assume now th a t d is not zero. Note th a t the induction on d allows us to ignore all the terms of degree less than d in equations (3.2.1),(3.2.2) and (3.2.4). But then the proof is the same as the one to the base case, since the equations obtained in the previous manner are the same as those for the case d = 0. 3.3.1

□

R em arks abou t th e algorithm

1. The algorithm of Thm. 3.17 provides in particular an algorithm to compute the 3-pointed Gromov-Witten coefficients for any homogeneous variety G /P . Different algorithms for th a t are discussed in §3.5 below. 2. The hypothesis (a) in the Theorem can be changed to (a’) The coefficient

is equal to 0 unless d = 0 when it is equal to 1.

and (a”) The equivariant quantum Chevalley terms

v, as given in the formula

(3.1.1). The proof goes the same way, except th a t in Step 1 the coefficient c^’f, is com­ puted starting from

w with s((3) -< w. Note th a t this requires less number of

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

43

computations, but one has to input the equivariant quantum Chevalley coefficients, which are known anyway from formula (3.1.1). 3. To reduce further the number of computations needed in the algorithm, one can also impose the following conditions: (i) c l i is equal to 0 if it has negative polynomial degree. is equal to 0 if c(u) + 1 > c(w) +

(ii)

^2

n(3dp or c{y) + 1 > c(w) + X) npdp by

Lemma 3.5. 4. The number of computations can be reduced even further than in previous remark, provided one knows some pure quantum coefficient of the form

with

u < w. As in Remark 2, this coefficient is then used in Step 1 to compute

.

3.4

C onsequences in equivariant cohom ology o f G /P .

A by-product of the previous algorithm (more precisely of the formulae (3.2.2) and (3.2.4)) is a formula for the equivariant coefficients of the form c™w = c™’°. Formulae for these coefficients were known before ([8]) therefore we obtain in particular some interesting combinatorial identities. If u

-ft

w formula (3.2.2) implies th a t c™ = 0. For u -< w formula (3.2.4) together

with the fact th a t c^o w = 1 shows th at (3.4.1)

c™w = 1/R (w 0, w, f )

for any assignment / (cf. Def. 3.13). If u -< w, but u is not equal to w, using the same formula yields (3.4.2)

c™w = R(u, w; / ' ) / R (w 0t w\ f )

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

44

for any assignments / and / '. We recall a different formula for the coefficient c™w: proved in [8] (see also Prop. 11.1.11 from [50])1. For

€ A, denote

by sij and let s%l■...■ slp be a reduced word decomposition

for the representative w'J in W p . For each j between 1 and p define (following [8] §4) (3.4.3)

rw(j) =

• ... • S i ^ p ^ )

P ro p o s itio n 3.19 ([8], Lemma 4.1, [50] Prop. 11.1.11). L e tu ,w be two representa­ tives in W p such that u < w . Then the equivariant coefficient c™w is equal to (3.4.4)

^ 2 r w(ji) ■ ■rw(jk)

where the sum is over all ordered sequences Ji < J 2 < ••• < jk such that Sih •... • s* is a reduced word decomposition for uy .

As a Corollary, we would like to note the following combinatorial properties of the function R ( u , w , f ) , which are not at all clear from its definition: C o ro lla ry 3.20. Let u ,w be two representatives such that u -< w. Then (a) R ( u , w ; f ) does not depend on the choice of the assignment f , i.e. R (u ,w \f) = R (u ,w ;f) for any two assignments f and f . Denote this function by R(u,u>). (b) R (u ,w ) is not equal to zero. Consequently, for any u -< w, the coefficient c f w is not equal to zero. (c) R ( u , w ) / R ( wq , w ) = Y ^ rw{ji) • ••• • rw{jk) where the sum is as in the equation (3.14)-___________________ 1The equivariant coefficient is equal to tp £“V(u)v) where ip sends the root a to wo(a) (see the paragraph before Prop. 2.5), and £^(10) is the coefficient considered in [8], Thm. 4.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

45

Proof. If u = w 0, both (a) and (b) follow from formulae (3.4.1) and (3.4.2). For general u such th a t u -< w, (a) follows from the fact th a t R (w 0,w; / ) does not depend on / , and th a t the quotient R(u, ui; f')/R (w o,w ; f ) does not depend on / and f , being equal to the equivariant coefficient c£w. For (b), by Lemma 3.14, there exists an assignment / such th a t R(u, w, f ) is not equal to zero, hence this is true for all assignments, by (a) just proved. Then, formula (3.4.2) implies th a t c™w must also be nonzero, (c) follows from Prop. 3.19.

□

Remark 3.21. The fact th a t c f w is not equal to zero if u -< w can also be derived from Prop. 3.19, or using the fact th a t c‘f w is equal to the localization of the Schubert class a(u) to the T —fixed point w P E G / P (cf. [3]). 3.5

A b rief survey o f th e algorithm s com p u tin g th e equivariant or quan­ tu m L ittlew ood-R ichardson coefficients

In this section we recall the previous algorithms from the literature to compute the quantum Littlewood-Richardson coefficients (i.e. the 3-point, genus 0, GromovW itten invariants), and compare them with the algorithm we propose, implied by the algorithm for the computation of EQLR coefficients. Note first th a t the quantum (or equivariant quantum) cohomology is not functorial, hence the Gromov-Witten invariants on G / P cannot be computed directly from those on G /B . However, there is a highly non-trivial result, conjectured by D. Peterson, and proved by C. Woodward [73], which shows th a t any Gromov-Witten invariant on G / P is equal to a certain such invariant on G /B , of possible different degree. This formula is known in the literature as Peterson’s comparison formula, and it is the main tool for the computation of the Gromov-Witten invariants on a general G /P .

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

46

The things are easier in type A (when G = PGL{m,)), and in some limited other cases (type B,C,D maximal Grassmannians), when Peterson’s comparison formula is not needed. In these cases, the algorithms are based on three ingredients: 1. A presentation of quantum cohomology, with generators and relations. 2. A “quantum Pieri” formula, which computes the multiplication of a quantum Schubert class with a generator from (1). 3. A “quantum Giambelli” formula, which gives a polynomial representative of a quantum Schubert class in terms of generators from ( 1 ). In type A, this program has been carried out first by A. Bertram [5] for Grassman­ nians, then in [20, 25] for full-flag manifolds and in [19] for partial flag manifolds. The ideas of Bertram and Ciocan-Fontanine from loc. cit. have been used by A. Kresch and H. Tamvakis in [49] in the case of Lagrangian and Orthogonal Grass­ mannians for the same purpose. Recently, A. Buch, using the notions of kernel and span of a rational curve (see also §5.2) recovered the quantum Pieri rules and Gi­ ambelli formulae in the case of Grassmannians ([13]), flag manifolds [15, 14] for the (maximal) Grassmannians of type B,C,D ([16]). In fact, using these ideas, A. Buch, A. Kresch and H. Tamvakis [16] have shown th a t the Gromov-Witten invariants in (type A, and types B,C,D - the maximal one) Grassmannian are in fact classical Littlewood-Richardson coefficients in 2-step flag manifolds. A different, more effective, algorithm for the Gromov-Witten invariants on the (type A) Grassmannian is the “rim-hook” algorithm of A. Bertram - I. CiocanFontanine - W. Fulton [6 ]. Related ideas also play a role in A. Postnikov’s definition of a family of symmetric functions, called toric Schur functions, which, when expanded in the basis of Schur functions, yield the Gromov-Witten invariants as coefficients

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

47

(see also [52]). We also mention a more geometric way of computing these invariants, using degenerations, which is due to I. Coskun [21]. The computation of the Gromov-Witten invariants for general X = G / P is far less developed. There is a quantum Chevalley formula (i.e. a multiplication with a divisor Schubert class) conjectured by D. Peterson and proved by W. Fulton and C. Woodward [28], but in general there is no presentation or quantum Giambelli formula. Moreover, the divisor classes do not generate the quantum cohomology, unless P = B. Nevertheless, one can use Peterson’s comparison formula to reduce the computation to the case when X = G /B . In this case there is a presentation, due to B. Kim [41] (see also [56]), with generators (quantum) divisor Schubert classes, and the Chevalley formula gives a multiplication of a quantum Schubert class by these generators. There is no explicit quantum Giambelli formula, but an algorithm for such a polynomial representing the quantum Schubert classes was obtained by L. Mare [57], using ideas from [25]. There is also a differential-geometric approach to the quantum cohomology of G /B , initiated by Guest [31], which was used recently to produce algorithms for the quantum Schubert polynomials and to recover the Chevalley formula (see also [2 , 55]). Our algorithm to compute the usual Gromov-Witten invariants on G /P , implied by the algorithm computing the EQLR coefficients, is conceptually simpler. It needs the equvariant version of the Gromov-Witten invariants, but just those invariants associated to G /P , hence avoiding Peterson’s comparison formula. It does not re­ quire the knowledge of a presentation or EQ Giambelli formula and it uses only a multiplication formula with divisor Schubert classes. Recall th at, unless P = B, the divisors do not generate the EQ (or quantum) cohomology algebras. Thus, in this sense, the EQ cohomology behaves better than the quantum one.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

48

3.6

A p p en d ix - P r o o f o f th e L em m a 3.11

The aim of this Appendix is to sketch the proofs of some of the properties of the equivariant coefficients c f ^ w, defined in §2.2. These properties are needed in the proof of the algorithm for the EQLR coefficients. We use the notations from §2.1. ■... • ,sik (where st] — s0i.) is a reduced word

Recall from formula (2.2.2) th a t if

decomposition for w € W , and /? = (f is a simple root in A then D(s0,w) = Y^Sh

i(/3)

ij=i

We show first th a t D (s0, w) ^ 0. For that, it is enough to show th a t each term of its defining sum is nonnegative (in the sense above). This follows immediately from the following Lemma: L em m a 3.22. Let w € W and a a positive root such that l(wsa) > l{w).

Then

w (a ) > 0. Proof. See the proposition in [37], §5.7. The second Lemma gives an equivalent definition of D(s0,w). Recall th a t

□ oj0

is

the fundamental weight corresponding to the positive simple root (3. Lem m a 3.23. The coefficient D (s0,w) is equal to D ( s p , W) = U J 0 - WU/3

Proof. See [50], Cor. 1.3.22. or Thm. 11.1.7(c).

□

P rop osition 3.24. Let u ,w be two distinct representatives in W p . There exists a positive simple root (3 in A \ A P such that D (s0,w) — D(s0, u ) is not equal to zero. Proof. Assume D (s0, w) — D (s0, u) for any (3 e A \ A p. Then Lemma 3.23 implies th a t w(ui0) = u(ujfj), i.e.

u

~ 1 w ( oj0 )

= oj0 for all (3 as before. Take p =

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

49

Then (p, (3) > 0 for any f3 € A \ A P and (p, (3) = 0 for any f3

6

Ap. Moreover,

u~1w(p) — p. Then by [10], Ch. 5, §4.6 (see also [37], Thm. from §1.12) it follows th a t u~1w must be in Wp, which contradicts the hypothesis.

□

The next Proposition shows th a t the difference D(sp,w) — D(sp, u), for (3 in A, satisfies a positivity property, provided th a t u is less than w in the B ruhat ordering. Recall (see e.g. [37], §5.9) th a t in this ordering, denoted ^ , u is less than w if there is a chain u = u0, u i , ..., Uk = w such th a t ui+1 = UiSa where a is a positive root such th a t l(ui+1) > l{ui). Note th a t u ^ w in the Bruhat ordering if and only if w

u in

the ordering defined in §3.2. P ro p o s itio n 3.25. Let u, w be two permutations in W such that u ^ w, and let (3 be a positive simple root. Then D(sp, w ) —D(sp, u ) ^ 0. Proof. First note th a t one can reduce to the case when w covers u, hence w — u sa for a a positive root. Then w{cop) = usa(wp) is equal to u(u>j3 - (ha,u>p)a) = u(up) - (ha,cop)u(a) hence (3.6.1)

D (sp,w) - D(sp, u) = (ha,ujp)u{a)

Then (ha,ujp) ^ 0 since a is a positive root and u(a) > 0 by Lemma 3.22, since l(usa ) > l{u).

□

P ro p o s itio n 3.26. Let u ,w be two distinct representatives in W p such that u ^ w in Bruhat ordering. Let Cov^(u,w) be the set of positive roots a in + \ that usa is a representative in W p , l(usa ) = l(u) + 1 and usa ^ w. Then (a) The set C ov^(u,w ) is nonempty.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

such

50

(b)

Let a be any positive root in Cov 0. Proof. By Prop, from §5.11 in [37] there exists a chain

U = Uo ^

U\ ^

... ^ Uk = w

in W , with k = l(w) —l(u) such th a t tti+ 1 = UiSat and l(ui+1 ) = l(ui) + 1, with a* in d>+. Modulo the Weyl group Wp of P this determines a chain in W /W p between uWp and wWp, necessarily of the same length k, since u, w are representatives in W p whose length difference is k. In particular, this shows th a t no a* can be in p and th a t the u f s are minimal length representatives for the cosets in W /W p , i.e. th a t uf1s are in W p . In particular, the positive root cto defined by u\ = uqs„u must be in Cov^(u,w ), which finishes the proof of (a). To prove (b), note th a t Prop. 3.25 implies th a t D(s0,u s a) — D (sp ,u ) > 0, hence ha{uj0)u{a) > 0 by formula (3.6.1). Lemma 3.22 implies th a t u(a) > 0 (because l(usa) > l(u)), so ha {ojfj) must be a nonzero, hence positive, integer, which ends the proof.

□

We interpret now these properties in the terms of the equivariant coefficients c4 P)w defined in Prop. 2.5. Recall th a t these are defined as ip(D(s0,w v )) where ip is the automorphism of $ sending the positive simple root /?* to the negative simple root w0((3i) (to see this note th a t wq(/?,) must be a negative root, by Lemma 3.22, and th a t sWo(0i) is equal to wos0iwo; the last perm utation sends all the positive roots but one to positive roots, hence l(wos0iwo) = 1). To state these properties, we recall from §3.2 the analogue of the set C ov^(u,w) for the ordering - [ A ] ,

where each fi G H f(p t) can be written as a linear combination of monomials in Pi, ..,,Pd with nonnegative integer coefficients. Remark: Graham ’s result deals with the more general situation of a variety Y , possi­ bly singular, a connected, solvable group B ' = T'U' acting on it and a T ' —stable subvariety V of Y . V determines only an equivariant homology class [V}t >€ H f dimV(Y), therefore his positivity result takes place in Hfi (Y). If Y is smooth the equivariant homology and cohomology are identified via equivariant Poincare duality ([12] §1), and one recovers Prop. 4.1 (for Y = X x X ). C o ro lla ry 4.2. Let X x X be endowed with the previous B ' —action, and V a T —stable subvariety. Then

\V]t

can be written uniquely as

[V} t = ' £ / M Y ( u ) x Y ( v )}t

in H f ( X x X ) where each fi is a polynomial i n x i,.. ., x r with nonnegative coefficients.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

54

Proof. Note first th a t the weights of T acting on Lie(U') (= Lie(U~) x Lie(U~)) are the same as the weights of T acting on Lie(U~), which are the negative roots of G. To finish the proof, it is enough to show th a t the B ' —stable subvarieties of X x X are precisely the products of Schubert varieties Y (u) x Y (v), and th a t they determine a basis for the equivariant cohomology of X x X . To do th at, recall th a t X has finitely many U~ —orbits, the Schubert cells Y(w)°. These orbits are B ~ —stable (they are B ~ —orbits of the T —fixed points)1, and they cover X with disjoint afRnes. Then the unipotent radical U' = U~ x U~ of B' acts on X x X with finitely many orbits Y{u)° x Y(v)° (u, v e W p ), and these orbits cover X x X by disjoint affines. Their closures are products of Schubert varieties Y (u ) x Y (v), and determine a basis [Y(u) x Y (v)\T for the equivariant cohomology H f ( X x X ) . Moreover, since any U'—orbit is B ' —stable, it follows th at any irreducible B ' —stable variety must be one of Y (u ) x Y ( v ) .

□

We state next a slightly more stronger version of the lemma 3.3, based on Kleiman’s transversality theorem, from the previous chapter. In this form, it is also stated (and proved) in [28] §7: L em m a 4.3 ([28], Lemma 7.2). Let Z be an irreducible G —variety and let F : Z — > X x X be a G —equivariant morphism, where G acts diagonally on X x X . Then, for any u and v in W p , the subscheme F ~ 1 (X(u) x Y (v)) is reduced, locally irreducible, of codimension c(u) + l(v). Given v

6

W p , we apply Lemma 4.3 to Z — Mo, 3 (X ,d ) (with the G —action

induced from X , as in §5), F = (ev3 ,ev3) (recall th a t ev 3 is the evaluation map, cf. §4) and u — Wo (the longest element in W p ). Mo, 3 {X,d) is irreducible by [42, 70] 1T h e fact th a t X has finitely m any o rb its im plies th a t any t / - —o rb it is B ~ —sta b le holds in a m ore general context, see [35], L em m a 3.3.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

55

and F is clearly G —equivariant. Then ev^1(Y(v)) = F ~ 1( X x Y (v)) is reduced, locally irreducible, of codimension l(v). Next result shows slightly more: L em m a 4.4. The inverse image ev^1(Y(v)) o f Y { y ) is a disjoint union of T —stable, reduced, irreducible components, each of them with codimension equal to l(v), the codimension o fY { v ) in X . Proof. It remains to show th a t each component Vi of e v f 1(Y(v)) (which is also a connected component) is T —stable. Since the whole preimage is T —stable, it follows th a t the T —action sends one connected component to another. But the identity id in T fixes all the V)’s so each component must be T —stable. 4.2

□

P r o o f o f th e p o s itiv ity T h e o re m

The idea of proof is to use the projection formula (A.l) and the push-forward formula (A. 2) to reduce the computation of the EQLR coefficients from an (equivari­ ant) intersection problem on a moduli space to an (equivariant) intersection problem on the product X x X , where we apply Corollary 4.2. If Y is a variety, denote by 7ry the structure morphism it : Y — > pt. Denote by M. the moduli space Ado,3 (A, d) and by ttm the morphism ttM03(x,d)T h e o re m 4.5. Let u, v, w in W p and d a multidegree. Then the equivariant quantum Littlewood-Richardson coefficient

is a polynomial in the (negative) simple roots

x i ,. .. ,x r with nonnegative coefficients. Proof. Let F : M. — >X x X be (ev1,ev2). Clearly, F is T —equivariant and proper. The main point of the proof is the following Claim:

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

56

Claim: The EQLR coefficient d f f is equal to 5 > (* M

)r([X (« ) X X(v)]T ■[F(V,)]r)

in H f(pt), where the sum is over the components V. of e v f 1(Y(w))

and a* is the

degree of the map F\vt : Vi —» FfVf) or 0 if dim V. > dimF(Vj). We postpone the proof of the Claim until later, and we show first how the Claim implies the theorem. Clearly, it is enough to show th a t each

ffiX x X ) l ( [ * ( « ) x X(v)}T • [F(V5)]t)

(4.2.1)

is nonnegative in the sense of the theorem. By Corollary 4.2, each [F{Vi)}T can be written as a combination ^ 2 fj[ Y ( u j) x Y (vj)\T, with Uj , Vj in W p , where each f is a nonnegative polynomial in variables aq, ...,xr. Projection formula (A .l) implies th a t the expression (4.2.1) can be w ritten as a combination £ ^ x v ) r ( [ X ( u ) x X(v)}T ■[Y(Uj) x l> ,.) ] r ) with each of Pf a nonnegative sum of / j ’s, hence also nonnegative. Write pr:l : X x X — >X for the projection to the i —th component of X (i = 1,2). Then (*rxxx £ ( [ * ( « ) X X (v)]T • \Y{Uj) x Y{vj)]T) can be w ritten as f f i x * x ) l ([* (« ) x X(v)}T • [Y(Uj) x Y ( Vj)]T)

=

f f i x x x ) l (Cp r fy f fi( u ) T) ■(pr£)*(it (v )t ) • [Y(Uj) x Y(vj)}T)

=

ffix )l{o { u )T ■[Y(uj-jjr) ' (*x)* (v (v )t ■[Y(vffir)

=

§UtUj • 5V)Vj

The first equality follows from the fact th a t [AT(u) x X ( v ) \T = (prf)*(a(u)T) • (prf y ( o ( v ) T), the second equality is Lemma 2.3 from [35] and the third equality

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

57

follows from equivariant Poincare duality (Prop. 2.2). This proves the positivity theorem.

□

Proof of the Claim. Recall (§5) th a t e f'f is defined by

((evi)* (a (u f ) ■ievl)*{a{v)T) ' (evJ Y ( (r(w )T)) Consider the composite (FT e v T )

X i x ----------* {X x X ^ x x X x — (X x x b t X x )

x

Xx

Then (e v ln ^u f) ■

) • (e v jy (5 (w f)

=

(F t )*([X(«) X X(v)}T) ■(ev[)*(a(w)T) But the composite

1 7

F

v

KxxX

M. ------ > X x X

.

> pt

is equal to 7Tj j , therefore c f f is equal to

0, and th a t A+ is included in the p x (to —p) rectangle only if Ai < to and A„ =

0.

L

__

_l

Figure 5.2: Example of partitions A and A

Write Av for the partition dual to A, i.e. the partition whose Young diagram is the rotation with 180 degrees of the complement of the Young diagram of A, in the given rectangle. A' will denote the partition conjugate to A, i.e. the partition in the (to —p) x p rectangle whose Young diagram is the transpose of the Young diagram of A. The zero partition (i.e. the partition with all parts of length zero) is denoted by (0). Note th a t •••, £m) = C m

where £* = (0,..., 1,..., 0), with

1

in the i —th position. Given the flag F „ define the

opposite flag, denoted F ° pp defined by FT* : (em) c ( £m, £m-l ) c ... C (em,e m_ 1 ,...,£ i) = C m.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

62

Let A = ( A i , A p) be a partition included in the p x (m — p) rectangle. One can define the Schubert variety with respect to the flag F, and partition A, denoted D.\(T»),

n x(F.) = {V e G r{p,m ) : dim (R fl

^ i}

If the flag F, is replaced by the opposite one, the expression above defines the opposite Schubert variety.1 Denote by ax and ax respectively the cohomology class in H ‘2^ ( G r ( p ,m ) ) deter­ mined by Q\(F.) and Q\(F°pp). It is well-known th a t the Schubert classes do not depend on the choice of the flag F,, so, in fact o \ = o \ (but the equivariant version of these classes will be different). As in the general case, the classes {co,} form a Z—basis for the integral cohomology of Gr(p, m) (see [26] P art III for an exposition about the subject). The multiplication in the cohomology ring is determined by the Littlewood-Richardson (LR) coefficients c-( , which are positive integers, counting the number of points in the intersection of the Schubert varieties VL\{F,), QM(G'.) and n vv(H,), where F,,G , and H, are three general flags and v y is the dual partition of v. The coefficients are 0 if |A| +

|/ i |

^ \v\. Geometrically,

where f x is the Gysin push-forward from §2.1. Positive combinatorial formulae for these coefficients, found in the literature as Littlewood-Richardson rules, are known (see e.g. [26] P art I or [69] and references therein). 5.1.3

Equivariant cohom ology

We discuss first a more concrete realization of A, the T —equivariant cohomology of a point. Recall from §2.3.2 th a t there is a direct system of T —bundles p : E T n — > 1In th e n o tatio n from §2.2, th e S chubert variety Q,\(F^PP) corresponds to K (u;v ), w here w is th e p erm u tatio n corresponding to A.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

63

BT n given by m

m

n (C ” \ { i=

0 })

— ►J J P

1

”' 1

i —1

approximating the universal T —bundle E T —*■B T , and that H'r(;pt) = H*T (BTn) for n large enough. It follows th a t A is the polynomial ring A = Z [ T i , Tm} where T t has complex degree 1, and it is equal to the first Chern class c i ( O r n - i ( l ) ) (•) of the line bundle 0 ( 1 ) on P ”^ 1, the i —th component of the product Y\T=i P"-1 We recall the definitions of the equivariant Schubert classes and the equvariant Poincare duality. The Schubert varieties Ct\(F,. ) and Q.X(F°PP) are T —stable (because the flags F. and F°pp are T —stable), so they determine equivariant cohomology classes crj and a f in H ^ ( X ) . The classes { 1. Assume th a t C =

(J

where

and C ® are trees of rational curves in C

intersecting in some point x £ C. Let rfW be the degree of / restricted to

Let

W (*), A"W be the span respectively the kernel of / restricted to C (t\ The induction hypothesis implies th a t d im (W ^) ^ p + d ^ \ and dim(ATb)) ^ p — (p l). But

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

and

65

C ^ intersect in a unique point x, and f ( x ) is a space of dimension p. Thus both and p+

contain the space f( x ) . It follows th a t dim (.span/) = dim(lT^b + W ^ ) F, + d ^ = p + d.

For the kernels, note th a t both K ^ and

are contained in /( # ) , with codi­

mensions at most dP:>respectively d!'2). Then the codimension of their intersection is + d ^ = d, which shows th a t dim(fcer/) = dim(iF^^ fl K ® ) > p —d.

at most

□

Denote by A(d) the partition obtained from A by removing its first d rows and by A(d) the partition obtained from A by removing the leftmost d columns. To be precise, if A = (Ai,...,Ap) then A(d) = (Xd+i, ■■■, \ p) while the i —th part (A(d))* of A(d) is equal to max(Aj —d, 0). Recall th a t ct(0) = 1. P ro p o sitio n 5.4. Let f : (C ,p i,p 2 ,pz) — > X be a stable map of degree d, let K be a (p — d)-dimensional subspace of the kernel of f and let W be a (p + d)dimensional subspace containing the span of f . For any complete flag F, : 0 C F\ C ... C Fm = Cm, if the image of f intersects Q\(F.) then K belongs to the Schubert variety Dx(d) (-^»)

Gr{p —d, m ) and W belongs to the Schubert variety &X(d) C^*) *n

Gr(p + d,m). Proof. The Proposition is Lemma 2 in [13] for C ~ IP1, but the proof for general C is the same. L e m m a 5.5. Let

□ be three partitions in the p x (m — p) rectangle such that

one of the following holds: 1. d < p and 2. d < m —p and a ^ Then

= 0 in H*{Gr{p — d, m)). = 0 in H*(Gr(p + d, m )).

= 0.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

66

To prove the Lemma we recall the following fact, proved in lemma 3.1:

Fact 1. Let F : X ' — ►Y be a T —equivariant morphism of two algebraic varieties, with Y smooth. Let V be a T —invariant subvariety of Y of codimension c and let \V ]t

€ H fc(Y) be the equivariant cohomology class of V. Then the equivariant co­

homology pull-back Ft([V ]t) is equal to 0 if F _1(V) is empty.

Proof of Lemma 5.5. The idea of proof to show th a t if

is different from 0 then

the intersection e v f 1(O a) in

LI e n J 1 ( f l A,) f l

e v f l (Llvv)

d) is nonempty. Then use Prop. 5.4 to get a contradiction. Assume th a t c%d is not 0. By the definition of the equivariant Gysin morphism,

the EQLR coefficient is equal to = ^ ( ( e t 'D V D

U ( « £ ) * ( < £ ) U ( e u f ) * ( 5 j v ))

Apply Fact 1 with X ' = A4o,3(X ,d), Y = X x X x X , V = Q\ x

x

and

F = (evi,ev 2 ,evf), where T acts diagonally on Y . Note th a t V is T —invariant and th a t the pull-back of its equivariant cohomology class in H ^(Y ) satisfies F f ( [ V ] T ) = « ) * ( a f ) U ( « £ ) > £ ) U (e u 3 r )* (^ v)

(this follows from the finite dimensional approximation approach, discussed in §2.2 above). Thus the inverse image i ?~1(V’), which is equal to the intersection e tq 1^ )

fl

e v f 1^ )

l~l

evf' i f l f)

must be nonempty in M.o,s(X, d). This amounts to the existence of a stable map / : (C,pi,p2,p3) — * X

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

67

whose image intersects Qa,^V and Q„v. Suppose p < d. Choose K f to be a (p — d)—dimensional subspace of the kernel of / (such a K f exists by Proposition 5.3). Proposition 5.4 implies th a t K f belongs to ^ \(d) ^

C Q^(d) ■ I11 particular the intersection flx(rf) ^

nonempty. B ut it

is a general fact th a t two Schubert varieties defined with respect to opposite flags are in general position (see e.g. [26], pag. 149) . It follows th a t the cohomology product °A(d) ' au7{d) must be nonzero in H*(Gr(p — d, to)), contradicting the hypothesis (1). The case when d < m —p is treated in a similar fashion, using a (p+d)—dimensional space W f including the span of / .

□

The key result of this paper is a sufficiently general condition th a t implies one of the hypothesis of Lemma 5.5, therefore giving a sufficient condition for the vanishing of the EQLR coefficients. This condition is spelled out in the Main Lemma. We divide its proof into two other lemmas, which we prove first, corresponding to the hypotheses (1) and (2) of Lemma 5.5. L em m a 5.6. Let A, p, v be three partitions included in the p x (m —p) rectangle and let d be a positive integer. Suppose that d < p and that |A| + d2 > \u\ + md. Then (d) ■°W{d) = 0 in H*(Gr(p - d, m)). Proof. To prove the Lemma, it is enough to verify the following inequality: |A(d)| + |^v(d)| > (p — d)(m —p + d) = dim (Gr(p —d, m)) Let A = (Ai,...,Ap) and v y = (pi,...,pp). Then A(d) — (A^+1, ..., Ap) and v y (d) = ( p d + i , ...,

pp).

The fact th a t |A| + d2 > |i^| + m d implies th a t cP + |A| + |^v | >

p(m —p) + md. Then d

p

d

i—1

i=d-\-1

j —1

p Pj > p(m — p) + m d — d2 j=d+1

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

68

hence d d y j (Ai + pi) > pirn - p) + m d - d2 ~ y \ i ~ y p j i=d+1 i=l j=1 p

But d y \ i=1

d + y p j < 2d(m - p) j =1

hence p

(A* + pi) > p(m — p) + md — d2 — 2d(m —p) = (p — d)(m —p + d) i—d-f-1 which finishes the proof of the Lemma.

□

L em m a 5.7. Let A, p, v be three partitions included in p x (rri —p) rectangle and let d > 0 be an integer. Suppose that d < m — p and that |A| + d2 > \v\ + md. Then CT\(d) ■H *(G r(m —p , m)) and it is well-known th a t ip* sends the class o \ € H*(Gr(p, m)) to the class a y £ H *(G r(m —p, m)), where A' is the partition conjugate to A. Note th a t the hypotheses of the Lemma 5.6 are satisfied when using partitions A ',//, v' in the (m —p) x p rectangle. Therefore

' aJPy(d) = 0 in H*(Gr(m —

p —d,m)). Using again the conjugation isomorphism for the Grassmannians G r(m — p — d,m ) and Gr(jp + d, m) gives th at

ax{d)'' a(Py7(d)' = 0 in H*(Gr(p + d,m )). Note th a t (z/)v = (^v)/- To finish the proof it is enough to prove the following combinatorial fact: F a ct: Let a, b, d be positive integers such th a t d < b and let A be a partition included in the a x b rectangle. Then (A'(d))' = A(d) in the (a + d) x (b — d) rectangle.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

69

Proof: Let A' = (£i,...,t&). Then (A(d))! = (td+i, ■■■■,%) — A'(d), which implies th a t a (d) = ((A (d))7 = ( m r -

□

Concluding, the two previous lemmas add up to: L em m a 5.8

(Main Lemma). Let A,/r, u be three partitions included in p x (m —p)

rectangle and

let d > 0 be an integer. Suppose that |A |+d2 > \v\ +md. Then

—0.

Proof of the Main Lemma: First note th a t d ^ p. Indeed, |A| + d2 > \v\ + m d and A C (m —p)p implies th a t p(m —p) + d2 > \v\ + md. p = d would imply d2 — d2 > \u\ which is impossible. It remains to study the cases when d < p and d > p. If d < p, apply Lemma 5.6 to get th a t

■ p, we claim th a t d < m — p. Indeed, since \uy \ = p(m — p) — \v\, we can rewrite the inequality |A| + \p\ ^ \v\ + m d as |A| + |/x| + \uv \ ^ p(m — p) + md But |A| ^ p(m — p) < d(m —p), hence d(m —p) + \fj\ + \vy \ > |A| + \pi\ + \vy \ ^ p(m — p) + md. Then \fi\ + \vy \ > p(m — p) + pd. Since \p\, \uy \ ^ p(m — p), one has th a t pd < p(m —p), which implies d < m — p, as needed. Hence we can apply Lemma 5.7 to get th a t a ^ d ,m )). Finally, Lemma 5.5, statem ent (2) gives th a t

= 0 in H*(Gr(p + = 0.

□

An immediate application of the Main Lemma is the next Corollary, which shows the vanishing of the mixed EQLR Pieri-Chevalley coefficients.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

70

C o ro lla ry 5.9. Let A, u be two partitions included in p x (m — p) rectangle and let d be a positive integer. Then

= 0, unless d = 1 and u = \ ~ .

Proof. If the polynomial degree of c f d^ is equal to 0, the assertion follows from the quantum Chevalley rule. Indeed, in this case, the quantum Chevalley rule from Prop. 2.6 translates to a \ * cr(i) = ^ 2 (l—>\

(5.2.1)

+ qax-

where the last term is ommited if A- does not exist. If the polynomial degree of c f d^ is positive, since the partition (1) is included in the d x d square, note th at |Aj + d2 ^ |A| + |(1)| > \v\ + md so the conclusion follows from the Main Lemma.

□

Buch’s methods imply an additional vanishing result, presented next: P ro p o s itio n 5.10. Let A be a partition included in the p x (m —p) rectangle. Then cS m -P)p = 0 f ° r d < min{p, m - p } . Proof. By Lemma 5.5, the result follows if (p —d)(m —p+d) — dim (Gr(p—d,m)) where the last inequality follows from the assumption d < m — p. 5.3

□

E q u iv a ria n t q u a n tu m C h e v a lle y -P ie ri ru le

We prove the equivariant quantum Chevalley-Pieri rule, then the recursive formula for the EQLR coefficients which is the main ingredient for the algorithm to compute them.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

71

One can use formula from Prop. 2.5 to get a formula for the equivariant coeffi­ cients

A or one can take the more direct formula from [46]. In any instance, the

equivariant coefficient cA^ is given by m

4 1

(5.3.1)

,(

)

=

E r *- E

r;

j= m -p + 1

* € /( A)

where 7 (A) is defined as follows: A traces a path starting from the NE corner of the p x (m —p ) rectangle and ending on the SW corner of the rectangle. Then 7(A) is defined by (5.3.2)

/(A) = {i : the i —th step of the path of A is vertical }.

For example, in the case p = 3 ,m = 7 and A = (4,2,1) (pictured below), 7(A) is equal to { 1 ,4 , 6 }.

Figure 5.3: The set of vertical steps of a partition: p = 3,m = 7 ,7(A) = {1,4,6}

We are ready to state the equivariant quantum Chevalley-Pieri rule: T h e o re m 5.11 (equivariant quantum Chevalley-Pieri rule). The following formula holds in Q H lf(X): (5.3.3)

ax o o - ( i ) =

+

1

c a> ( ) c t a

+ qcrx-

fi—*x where the last term is omitted if A- does not exist.

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

72

Proof. The equivariant quantum Pieri-Chevalley rule is a deformation of both equivari­ ant and quantum Pieri-Chevalley rules, so it must contain at least the (pure) equivari­ ant and quantum terms. It remained to prove th a t there are not any other terms. The only possibilities are terms of the form c f d^ a u where d > 0 and the polynomial degree of

is positive. But Corollary 5.9 shows th a t in this case

claimed.

= 0, as □

The equivariant quantum Pieri-Chevalley rule, commutativity and an associativity relation implies a quantum generalization of an equation satisfied by the equivariant LR coefficients (see e.g. Thm. 3 in [46] or [64, 65]). It relates the EQLR coefficient
H 2m- \ M , M \ X ) If U is an open set in X , there is a map H f M(X ) — > H BM(U) coming form the restriction H b m (X ) = H 2m- \ M , M \ X ) — + H 2m~i (M°, M° \ U) = H BM(U) where M ° — M \ Y and Y = X \ U. This map fits in a long exact sequence: (A.6)

... -► H b m (Y) -* H f M{X)

H b m (U) - f

-> ...

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

which is functorial with respect to proper maps / : X ' — ►X (see [26], page 218). The variety X has a fundamental class [X] in H ^ ^ m^ { X ) . Moreover, given V a subvariety of X , it determines a class r/y = **([V]) in H ^ m(y^{X) where i is the inclusion of V in X . If X is smooth, H f M(X ) = H 2dun(x )-*(x) and rjy is the class determined by V in the cohomology of X . We are now ready to prove Prop. A.5. Proof of Prop. A . 5. Write the morphism / as the composite

♦ Y where g is a closed embedding and pr is the first projection. If / is the projection pr (so X = Y x IP" and Y are smooth, and so are X ' — Y ' x P" and Y ') the result follows from the commutativity of the following diagram:

where N (resp. N ') denotes the normal bundle of Y ' in Y (resp. the normal bundle of X ' in X)] c denotes the codimension of X ' in X (equal to the codimension of Y ' in Y). Note th a t capping with \Y'\ resp. [X') is just the multiplication with the Thom class of the normal bundle N resp. N ' . Therefore the left diagram is commutative by the naturality of the Thom isomorphism, while the the right one is commutative by the functoriality of the long exact sequence (A. 6). If / is the closed embedding g the result follows by passing to the limit (using Cech cohomology) in the following commutative diagram: H ^ U ) x H 2d(U, U \ X )

—

resxres

H ^ U ') x H 2d( U ', U '^ X ') —

H 2d+i(U, U \ X ) re s

H 2d+i(Ul, U' \ X ')

JJ2d+i( y ) res

H 2d+i(Y')

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

100

where U is an open neighborhood of X in Y , U' = U D Y ' .

The regularity of

i is used to show th a t the fundamental class [X] G H 2d( Y , Y \ X ) restricts to \X'] e H 2d{ Y ' x ^ X ').

□

One can also define a cohomology Gysin map associated to a regular embedding i : X —>Y, but Y not necessarily smooth. To any such embedding one can associate an orientation u x y € H 2c( Y , Y \ X ) , where c is the (complex) codimension of X in Y . For a general definition of u x ,y , see [27], Ch. 19, §2. We only need the following situation: X (A.7)

—

Y f

r X'

Y'

where the square is a fiber square, / (and so / ') is flat, and X ' and Y ' is smooth. In this case i and j are regular embeddings of the same codimension and u x ' y G H 2c( Y ',Y ' \ X ') is the Thom class associated to the embedding j . Then ux,y — f*(ux,y)D efinition A .6 . W ith the notation above, define i* : H l( X) —> H't+2c(Y) to be given by composition: W ( X ) H 2c(Y, Y \ X ) ^ H i+2c(Y,

H i+2c{Y)

where U is given by the multiplication with the orientation class

u x

,y ,

as in A.5.

Note th a t when Y is smooth the two push-forward notions considered in this section agree. P rop osition A .7 (Properties of **). (i) Projection formula: i*{i*{y) U x) = y Ui*(ar)

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

101

in H*(Y). (ii) Given the fiber square A. 7,

as maps H fiX ')

H i+2e(Y).

(in) Consider the fiber square A .3 from Prop. A . 5 but where f and f

are now

regular embeddings of the same codimension, say d . Then f'S = ff* as maps H \ X ) -* H i+2c{Y'). Proof. To prove (i), we use the fact th a t the Cech cohomology and the usual coho­ mology coincide for the spaces considered. Let U be an open subset of Y , containing X . By excision H*(Y, Y \ X ) ~ H*(U, U \ X ). The following diagram is clearly commutative: H*(Y) ® H i(U) ® H 2c(Y, Y \ X )

—=-» (H fiY ) ® W { U ) ) ® H 2c{Y, Y \ X )

id

U

H fiY ) ® (W { U ) ® H 2c{Y, Y \ X ) )

H i+j - 2c(Y, Y \ X )

Then the right hand side of the projection formula is obtained by passing to the limit and restricting from H t+:>~2c(Y, Y \ X ) to fT +J_2c(Y) on the top and right sides of the diagram, while the left hand side is obtained similarly, using now the left and bottom sides of the diagram, (ii) follows using again Cech cohomology and the fact th a t the orientation

u x ,y

is the pull-back of the orientation

(hi) is the second

part (when / is the regular closed embedding) of the proof of Prop. A.5.

□

N o ta tio n A .8 . The cohomology push-forward irfix) (x € H*( X)) associated to 7r :

X —> pt will be denoted by Jx x. In certain situations, f x x will denote the

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

102

homology push-forward 7r*(x n [X]) (where [X] is the fundamental class of X ). We will make note when we use the latter notation. A .2

F in ite dim en sion al approxim ations and equivariant G ysin m aps

The topological spaces E T and B T are infinite dimensional, so in particular they are not algebraic varieties. Nevertheless, one can consider the direct system of finite­ dimensional T —bundles p : E T n — >B T n given by

II(C'*+1 ^ {0}) — . J j l i= l

i= 1

where r = dim T. The ordering on the bundles is given by inclusion (for the con­ struction above or similar ones see e.g. [12], [39], Ch.4 §11, or Ch. 7, [23] §3.1). Let X t ,u

E T n x T X be the induced finite dimensional approximations

of X T-Then

one can show th a t H ^ ( X ) is equal to H l (X Tj7l) for n large ([12]). Let / : X —> Y be a T —equivariant map of projective varieties, and let d = dim X —dim T . We define next an equivariant analogue f? : H ^X) — >

of the Gysin maps defined in §A. Assume th a t either Y is smooth or th a t / is a regular embedding. Then / determines a Gysin map of the finite dimensional approximations

:7T(xT,n) —»fr-M(yT,n). D efinition A .9. Define the equivariant Gysin map / J as the unique map th a t makes the following diagram commute: ^ ( X T>„) /*.n H i~2d(YTtn)

fT H i f 2d(Y)

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

103

for every integer n. The horizontal maps res are the cohomology pull-backs induced by the inclusions X r ,n

X T respectively YT Yp,n follows by applying Prop. A.5 if Y is smooth respectively Prop. A.7 (iii) if / is a regular embedding to the fiber square XT fn2

/ n x

Yt for integers rq
H T %+2c{Y). (iv) (Compatibility with non-equivariant Gysin maps) Consider the diagram below, induced by the map f : X —* Y , Y —smooth. H^X) (A.4)

f, ffi-2d(Y)

WT (X) fj H f f 2d(Y)

Then res o f f — /* o res as maps Hlp{X) —>/T ~ 2(dimX~dimy)(X ). Proof. The first three assertions follow from the corresponding ones made using finite dimensional approximations. The latter are proved in §A (Propositions A.3, A.4 and

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

105

A.7). (iv) follows by applying Proposition A.5 to the fiber square X

X-T , n fn

Y

-* y TiTI

□ N o ta tio n A . l l . For x £ H t (X), I Xt x denotes n j (x). As in the non-equivariant case, in certain situations, which will be noted when used, Jx x will also denote the equivariant homology push-forward 7rf (x D [AT]).

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

106

A P P E N D IX B

E q u ivarian t q u a n tu m c o h o m o lo g y - g en era l d e fin itio n and p ro o fs o f its p r o p e r tie s

The goal of this Appendix is to define the equivariant quantum cohomology, follow­ ing the ideas from [40]. The proof of the associativity of the equivariant quantum cohomology is the same as in the non-equivariant case, but using now equivariant versions of all the maps and facts involved. The main sources of inspiration are [1] and [71]. A ss u m p tio n B .l . We use the same assumptions on X as in [1]: 1.

X is a smooth, projective, T —variety, which is convex, i.e. for any map

f : F ' ^ X , H 1( F \ f * T x ) = 0; 2. The Chow and singular cohomology of X are isomorphic; 3. The effective classes in H 2(X ; Z) are precisely the nonnegative linear combi­ nations of curve classes of the form /* [P1]. Let e rf,..., o f be a A—basis for H*r {X) (such a basis exists by equivariant formality of X ([33], Thm. 14.1), which is implied by assumption 2 above). The T-action on X induces the diagonal action on AT x X . Let A be the image of X under the diagonal embedding, and let [A]y 6 H f dimX( X x X ) be the cohomology class determined by A; pi and p2 will denote the projections from X x X to X . The products o f o f

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

107

determine a basis for H ^ ( X x X ) ~ H^ { X ) ®a H^ ( X) , and the class [A]r can be expressed as

e,/ where ge^ are homogeneous polynomials in A. Then the equivariant multiplication o 1f U oJf of two basis elements can be w ritten as:

(B.l)

where, as usual, f x j j i —2 d i m X

denotes the equivariant Gysin push-forward n f : ffir ( X) —►

{pt).

We will need a slightly more general moduli space than M 0>3(X ,d), by allowing more marked points. Fix d E H 2( X]h). Let M 0tn(X, d) be Kontsevich’s moduli space of stable maps to X of degree d with n marked points. Its closed points are stable maps / : (C^pi, ...,pn)

X , denoted (C,pi, ...,pn]/ ) , where C is a tree of F x’s

with n smooth points marked on C and /» [C\ represents d. The stability of / means th a t every component of C which is contracted through / should contain at least three points which are either marked or intersection points with other components. W ith this notation, A40ln(X, d) is a projective variety of dimension dim Ado,n(X, d) = dim X + where f x denotes the homology push-forward and T X is the tangent bundle of X . As usual, there are evaluation maps evi : M 0>n(X ,d ) -» X

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

108

sending ( C , p i , pn; / ) to f(pi). There is also a contraction map fo r : M 0tn(X, d)

M 0,n

to Deligne-Mumford moduli space of stable curves, which “forgets the map / ” , send­ ing (C,pi, ...,pn', / ) to ( C , p i , ...,pn), where C is obtained from C by contracting the unstable components of C (cf. [71]). D efinition B .2. Let d be a homology class in H 2(X] Z) and 7i,...,7„ in H f ( X ) . The equivariant Gromov-Witten invariant f

By definition,

■■■Gin) is defined to be

(enf)*( 7 i) U (evf )*(j2) U ... U (et£)*( 7 „).

7 n )

is a homogeneous polynomial in A of (complex) de­

gree deg(7 1 ) + ... + deg( 7 „) —dim Ado,n(X, d). Note th at, with this definition, the equivariant multiplication erf U a j can be written as (B.2)

of U of =

o-f, v f ) 9 eJt f -

Fix o f , ... ,o f a basis of H f dlmX~2( X) (i.e of curve classes). Since the Chow and the usual cohomology of X are isomorphic, there is an isomorphism H!f(X)/A~H*(X) (again, by the equivariant formality of X , see [33]). Let ct* be the image of o f under this isomorphism. Then the classes o\, ...,os form a basis for H 2dimX~2(X), so each (effective) degree d can be written as d = fT'^diOj. Identify d with the sequence of nonnegative integers (d1; ...,d3). As usual the zero degree d = (0 ,..., 0 ) is denoted simply by 0 .

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

109

Next we prove some properties of the equivariant Gromov-Witten invariants, needed to prove th a t the equivariant quantum multiplication has unit and it is a deformation of the quantum one. P rop osition B .3 (Properties of the equivariant Gromov-Witten invariants). Let d be a degree, n ^ 3 a positive integer and j i , ..., 7 „ cohomology classes in H f ( X ) . The following hold: 1. Assume that the polynomial Id{7i, ■■■Tin) has degree 0. Then

Id (7 i. - , 7 n ) =

I

JMo,n(X,d)

e u * ( 7 i) U ... U e < ( 7 „ )

where % is the image o f ’ji under the isomorphism H ^ ( X )/A —> H*(X). 2. Let 1 be the unit in Hrf(X). Then

if d AO or n > 3;

{ 0, f x T 72 U 7 3 ,

ifn =

2>

and d = 0.

Proof. The first point follows from the commutativity of the diagram H 2D(AAo,s(X, d))

H 2D( M 0l3( X ,d ))

H 0(pt)

Hj-(pt)

where D denotes the dimension of the moduli space Mo, n(X, d) (cf. equation A.4). For the second point, consider first th a t d is not equal to zero. Consider the contraction map 7r. AA.Qn(fX,d)

>Ado,7i—i(A) d).

This map is clearly T —equivariant, and induces a map

T r , m ■ -A do,r i ( . X , d ) T , m

* -^d o ,7 i—1 ( - ^ j d ) T , m

R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

110

on the finite-dimensional approximations of the spaces involved. Then ^d(l) 7 2 )

•••? 7 r a ) =

_

( ^ T )m ) 4 ^ 0 , r i ( ^ , r f ) T , m ] n ( ( e n 2 ) r im ( 7 2 ) U - " U ( e v rl ) T )m ( 7 re'

J Mo,n-l{X,d)T,m

for m large enough, where ((7r)r,m)* and the integral are homology push-forward maps and 7 ^ is the restriction of 7 ,; from H f { X ) to 77*(A/r,m). Since (^ T ^ m )*

it follows th a t 7d(l, 7 2 ,..., 7 „) =

0

[Afo,n(A", d)^^]

0

, as desired.

Assume now th a t d = 0. Then M. 0 n[X, d) is isomorphic to X x M 0^n. If n > 3, a similar computation as in the case d ^

0

shows th a t 7^(1,7 2 , •••, 7 «) is equal to zero.

The case n — 3 and 7 = 0 is obvious.

□

i ^ s, consider an indeterminate qi with (complex)

For each curve class crf, 1 degree the nonnegative integer

deg{qi)= [