Chern class formulas for ₂ Schubert loci

We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, clarifying a previous computation of Edidin and Graham.


e G2 flag var
ety and its Schubert varieties, and several computations, including one that answers a question of W. Graham.In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in [Ed-Gr].Contents 1. Introduction 1 2. Overview 4 3. Octonions and compatible forms 9 4. Topology of G 2 flags 15 5. Cohomology of flag bundles 18 6.Divided difference operators and Chern class formulas 21 7. Variations 23 Appendix A. Lie theory 25 Appendix B. Integral Chow rings of quadric bundles 30 References 32

Introduction

Let V be an n-dimensional vector space.The flag variety F l(V ) parametrizes all complete flags in V , i.e., saturated chains of subspaces
E • = (E 1 ⊂ E 2 ⊂ • • • ⊂ E n = V ) (with dim E i = i).
Fixing a flag F • allows one to define Schubert varieties in F l(V ) as the loci of flags satisfying certain incidence conditions with F • ; there is one such Schubert variety for each permutation of {1, . . ., n}.This generalizes naturally to the case where V is a vector bundle and F • is a flag of subbundles.Here one has a flag bundle Fl(V ) over the base variety, whose fibers are flag varieties, with Schubert loci defined similarly by incidence conditions.Formulas for the cohomology classes of these Schubert loci, as polynomials in the Chern classes of the bundles involved, include the classical Thom-Porteous-Giambelli and Kempf-Laksov formulas (see [Fu1]).

The above situation is "type A," in the sense that F l(V ) is isomorphic to the homogeneous space SL n /B (with the subgroup of upper-triangular matrices).There are straightforward generalizations to the other classical types (B, C, D): here the vector bundle V is equipped with a symplectic or nondegenerate symmetric bilinear form, and the flags are required to be isotropic with respect to the given form.Schubert loci are efined as before, with one for each element of the corresponding Weyl group.The problem of finding formulas for their cohomology classes has been studied by Harris-Tu [Ha-Tu], Józefiak-Lascoux-Pragacz [Jó-La-Pr], and Fulton [Fu2,Fu3], among others.

One is naturally led to consider the analogous problem in the five remaining Lie types.In exceptional types, however, it is not so obvious how the Lie-theoretic geometry of G/B generalizes to the setting of vector bundles in algebraic geometry.The primary goal of this article is to carry this out for type G 2 .

To give a better idea of the difference between classical and exceptional types, let us describe the classical problem in slightly more detail.The flag bundles are the universal cases of general degeneracy locus problems in algebraic geometry.Specifically, let V be a vector bundle of rank n on a variety X, and let ϕ : V ⊗ V → k be a symplectic or nondegenerate symmetric bilinear form (or the zero form).If E • and F • are general flags of isotropic subbundles of V , the problem is to find formulas in H * X for the degeneracy locus D w = {x ∈ X | dim(F p (x) ∩ E q (x)) ≥ r w w 0 (q, p)}, in terms of the Chern classes of the line bundles E q /E q−1 and F p /F p−1 , for all p and q. (Here w is an element of the Weyl group, considered as a permutation via an embedding in the symmetric group S n ; w 0 is the longest element, corresponding to the permutation n n − 1 • • • 1; and r w (q, p) = #{i ≤ | w(i) ≤ p} is a nonnegative integer depending on w, p, and q.)Such formul s have a wide range of applications: for example, they appear in the theory of special divisors and variation of Hodge structure on curves in algebraic geometry [Ha-Tu, Pa-Pr], and they are used to study s k of Fehér and Rimányi, e.g., [Fe-Ri]).They are also of interest in combinatorics (e.g., work of Lascoux-Schützenberger, Fomin-Kirillov, Pragacz, Kresch-Tamvakis).See [Fu-Pr] for a more detailed account of the history.

In this article, we pose and solve the corresponding problem in type G 2 :

Let V → X be a vector bundle of rank 7, equipped with a nondegenerate alternating trilinear form γ : 3 V → L, for a line bundle L. Let E • and F • be general flags of γ-isotropic subbundles of V , and let
D w = {x ∈ X | dim(F p (x) ∩ E q (x)) ≥ r w w 0 (q, p)},
where w is an element of the Weyl group for G 2 (the dihedral group with 12 elements).Find a formula for [D w ] in H * X, in terms of the Chern classes of the bundles involved.

The meaning of "nondegenerate" and "γ-isotropic" will be explained below ( § §2.1-2.2), as will the precise definition of D w ( §2.5).In order to establish the relation between group theory and geometry, we give descriptions of the G 2 flag variety and its Schubert subvarieties which appear to be new, although they will not surprise the experts ( §4, §A.4).This is done in such a wa as to make the transition to flag bundles natural.We then give presentations of the cohomology rings of these flag bundles, including ones with intege coefficients (Theorem 5.4).Finally, we prove formulas for the classes of Schubert varieties in flag bundles ( §6); the formulas themselves are given in [An1,Appendix D.2].We also discuss alternative formulas, answer a question of William Graham about the integrality of a certain rational cohomology class, and prove a result giving restrictions on candidates for "G 2 Schubert polynomials" ( §7).

We also need a result on the integral cohomology of quadric bundles, which were studied in [Ed-Gr].Appendix B corrects a small error in that article.

Various constructions of exceptional-type flag varieties h ve been given using techniques from algebra and representation theory; those appearing in [La-Ma], [Il-Ma], and [Ga] have a similar flavor to the one presented here.A key feature of our description is that the data parametrized by the G 2 flag variety naturally determine a complete flag in a 7-dimensional vector space, much as isotropic flags in classical types determine complete flags by taking orthogonal complements.The fundamental facts that make this work are Proposition 2.2 and its cousins, Corollary 3.12 and Propositions 3.15 and 3.16.

Formulas for degeneracy loci are closely related to Giambelli formulas for equivariant classes of Schubert varieties in the equivariant cohomology of the corresponding flag variety.We will usually use the language of d generacy loci, but we discuss the connection with equivariant cohomology in §2.6.In brief, the two perspectives are equivalent when det V and L are trivial line bundles.

Another notion of degeneracy loci is often useful, where one is given a map of vector bundles ϕ : E → F on X, possibly possessing some kind of symmetry, and one is interested in the locus where ϕ drops rank.This is the situation considered in [Ha-Tu], for example, with F = E * and symmetric or skew-symmetric maps.We investigate the G 2 analogue of this problem in [An2].

When the base X is a point, so V is a vector space and the flag bundle is just the flag variety G/B, most of the results have been known for some time; essentially everything can be done using the general tools of Lie theory.For example, a presentation of H * (G/B, Z) was given by Bott and Samelson [Bo-Sa], and (different) formulas for Schubert classes in H * (G/B, Q) appear in [BGG].Since this article a so aims to present a concrete, unified perspective on the G 2 flag variety, accessible to general algebraic geometers, we wish to emphasize geometry over Lie theory: we are describing a geometric situation from which type-G 2 groups arise naturally.Reflecting this perspective, we postpone the Lie-and representation-theoretic arguments to Appendix A. We shall use some of the notation and results of this appendix throughout the article, though, so the reader less familiar with Lie theory is advised to skim at least §A.1, §A.3, and §A.5.

Notation and conventions.
nless otherwise indicated, the base field k will have characteristic not 2 and be algebraically closed (although a quadratic extension of the prime field usually suffices).When char(k) = 2, several of our definitions and results about forms and octonions break down.However, most of the other main results hold in arbitrary characteristic, including the description of the G 2 flag variety and its cohomology, the degeneracy locus formulas, and the parametrizations of Schubert cells; see [An1,Chapter 6] for details in characteristic 2.

Angle brackets denote the span of enclosed vectors: x, y, z := span{x, y, z}.

For a vector bundle V on X and a point x ∈ X, V (x) denotes the iber over x.If X → Y is a morphism and V is a vector bundle on Y , we will often write V for the vector bundle pulled back to X.If V is a vector space and E is a subspace, [E] denotes the corresponding point in an appropriate Grassmannian.

We generally use the notation and language of (singular) cohomology, but this

ould be r
ad as Chow cohomology for ground fields other than C. (Since the varieties whose cohomology we compute are rational homogeneous spaces or fibered in homo eneous spaces, the distinction is not significant.)

Acknowledgements.It is a pleasure to thank William Fulton for his encouragement in this project and careful readings of earlier drafts.Conversations and correspondence with many people have benefitted me; in p escription of the G 2 flag variety and statements of the main results.Proofs s.

2.1.Compatible forms.Let V be a k-vector space.Let β be a nondegenerate symmetric bilinear form on V , and let γ be an alternating trilinear form, i e., γ : 3 V → k.Write v → v † for the isomorphism V → V * defined by β, and ϕ → ϕ † for the inverse map V * → V .(Explcitly, these are defined by v † (u) = β(v, u) and ϕ(u) = (ϕ † , u) for any u ∈ V .)Our constructions are based on the following definitions:
Definition 2.1. Call the forms γ and β compatible if 2 γ(u, v, γ(u, v, •) † ) = β(u, u)β(v, v) − β(u, v) 2 (2.1)
for all u, v ∈ V .An alternating trilinear form γ : 3 V → k is nondegenerate if there exists a compatible nondegenerate symmetric bilinear form on V .

The meaning of the strange-looking relation (2.1) will be explained in §3; see Proposition 3.3.(The factor of 2 is due to our convention that a quadratic norm and corresponding bilinear form are related by β(u, u) = 2 N (u).)A pair of compatible forms is equivalent to a composition algebra structure on k ⊕ V (see §3).Since a composition algebra must have dimension 1, 2, 4, or 8 over k (by Hurw tz's theorem), it follows that nondegenerate trilinear forms exist only when V has dimension 1, 3, or 7.In each case, there is an open dense GL(V )orbit in 3 V * consisting of nondegenerate forms.When dim V = 1, the only alternating trilinear form is zero, and any nonzero bilinear form is compatible with it.When dim V = 3, an alternating trilinear form is a scalar multiple of the determinant, and given a nondegenerate bilinear form, it is easy to show that there is a unique compatible trilinear form up to sign.

When L(V )-orbit, especially if char(k) = 3, but it is still true (Proposition A.1).The choice of γ determines β uniquely up to scalar -in fact, up to a cube root of unity (see Proposition A.3).

Associated to any alternating trilinear form γ on a seven-dimensional vector space V , there is a canonical map B γ : Sym 2 V → 7 V * , determining (up to scalar) a bilinear form β γ .We will give the formula for char(k = 3 here.Following Bryant [Br], we define B γ by
B γ (u, v) = − 1 3 γ(u, •, •) ∧ γ(v, •, •) ∧ γ, (2.2)
where γ(u, •, •) : 2 V → k is obtained by contracting γ with u.Choosing an isomorphism 7 V * ∼ = k yields a symmetric bilinear form β γ .If β γ is nondegenerate, then a scalar multiple of it is compatible with the trilinear form γ; thus γ is nondegenerate if and only if β γ is nondegenerate.The form β γ is defined in characteristic 3, as well, and the statement still holds (see Lemma 3.9 and its proof).


Isotropic spaces.

For the rest of this section, assume dim V = 7.Given a nondeg nerate trilinear form γ on V , say a subspace F of dimension at least ace is γ-isotropic if it is contained in a 2-dimensional γ-isotropic space.If β is a compatible bilinear form, every γ-isotropic subspace is also β-isotro egenerate, a maximal β-isotropic subspace has dimension 3. Proposition 2.2.For any (nonzero) isotropic vector u ∈ V , the space
E u = {v | u, v is γ-isotropic} is three-dimension l and β-isotropic. Moreover, every two-dimensional γ-isotropic subspace of E u contains u.
The proof is given at the end of §3.2.The proposition impl es that a maximal γ-isotropic subspace has dimension 2, and motivates the central definition:
Definition 2.3. A γ-isotropic flag (or G 2 flag) in V is a chain F 1 ⊂ F 2 ⊂ V
of γ-isotropic subspaces, of dimensions 1 and 2. The variety parametrizing γisotropic flags is called the γ-isotropic flag variety (or G 2 flag variety), and denoted F l γ (V ).

The γ-isotropic flag variety is a smooth, six-dimensional projective variety (Proposition 4.1).See §A.4 for its desc iption as a homogeneous space.

Proposition 2.2 shows that a γ-isotropic flag has a uni o a complete flag in V : set F 3 = E u for u spanning F 1 , F ⊥ i , with respect to a compatible form β. (Since a compatible form is unique up t

scala
, this is independent of the choice of β.)This defines a closed immersion F l γ (V ) ֒→ F l β (V ) ⊂ F l(V ), where F l β (V ) and F l(V ) are the (classical) type B and type A flag varieties, respectively.

From the definition, there is a tautological sequence of vector bundles on F l γ (V ),
S 1 ⊂ S 2 ⊂ V,
and this extends to a complete γ-isotropic flag bundles Q i = V /S 7−i .

2.3.Bundles.Now let V → X be a vector bundle of rank 7, and let L be a line bundle on X.An alternating trilinear form γ : 3 V → L is nondegenerate if it is locally nondegenerate on fibers.Equivalently, we may define the Bryant form B γ : Sym 2 V → det V * ⊗ L ⊗3 b Equation (2.2), and γ is nondegenerate if and only if B γ is (so
B γ defines an isomorphism V ∼ = V * ⊗ det V * ⊗ L ⊗3 ). A subbundle F of V is γ-isotropic if each fiber F (x) is γ-isotropic in V (x); fo F of rank 2, this is equivalent to requiring that the induced map F ⊗F → V * ⊗L be zero. If F 1 ⊂ V is γ-isotropic, the bundle E F 1 = ker(V → F * 1 ⊗ V * ⊗ L) has rank 3 and is isotropic for B γ . (If u is a vector in a fiber F 1 (x), then E F 1 (x) = E u , in the notation of §2.2.)
Given a nondegenerate form γ on V , there is a γ-isotropic flag bundle Fl γ (V ) → X, with fibers F l γ (V ( c subbundles S i and quotient bundles Q i , as before.

2.4.Chern class fo the Weyl group.There is an embedding of W = W (G 2 ) in the symmetric group S 7 such that the p rmutation corresponding to w ∈ W is determined by its first two values.We identify w with w(i) ≤ p}.

(2.3) Given a fixed γ-isotropic flag F • on X, the Schubert loci are defined by
Ω w = {x ∈ Fl γ (V ) | rk(F p → Q q ) ≤ r w (q, p) for 1 ≤ p ≤ 7, 1 ≤ q ≤ 2}.
These are locally trivial fiber bundles, whose fibers are Schubert varieties in F l γ (V (x)).

The G 2 divided difference operators ∂ s and ∂ t act on Λ[x 1 , x 2 ], for any ring Λ, by
∂ s (f ) = f (x 1 , x 2 ) − f (x 2 , x 1 ) x 1 − x 2 ; (2. 2 ) − f (x 1 , x 1 − x 2 ) −x 1 + 2x 2 . (2.5) If w ∈ W has reduced word w = s 1 • s 2 • • • s ℓ (
where s i is the simple reflection s or t), then define ∂ w to be the composition
∂ s 1 • • • • • ∂ s ℓ .
This is independent of the choice of word; see §A.5.(As mentioned in §A.3, each w ∈ W (G 2 ) has a unique reduced word, with the exception of w 0 , so independence of choice is actually la case.)These formulas also define operators on
H * Fl γ (V ). (See §6.)
Let V be a vector bundle of rank 7 on X equipped with a nondegenerate form γ : 3 V → k X , and assume det
V is trivial. Let F 1 ⊂ F 2 ⊂ • • • ⊂ V be a complete γ-isotropic flag in V . Set y 1 = c 1 (F ), y 2 = c 1 (F 2 /F 1 ). Let
Fl γ (V ) → X be the flag bundle, and set x 1 = −c 1 (S 1 ) and x 2 = −c 1 (S 2 /S 1 ), where S 1 ⊂ S 2 ⊂ V are the tauto ogical bundles.

Theorem 2.4.We have [Ω w ] = P w (x; y), where P w = ∂ w 0 w −1 P w 0 , and
P w 0 (x; y) = 1 2 (x 3 1 − 2 x 2 1 y 1 + x 1 y 2 1 − x 1 y 2 2 + x 1 y 1 y 2 − y 2 1 y 2 + y 1 y 2 2 ) ×(x 2 1 + x 1 y 1 + y 1 y 2 − y 2 2 )(x 2 − x 1 − y 2 ). in H * (Fl γ (V ), Z).
(Here w 0 is the longest element of the Weyl group.)

The proof is given in §6, along with a discussion of alternative formulas, including ones where γ takes values in M ⊗ ine bundle M .2.5.Degeneracy loci.Returning to the problem posed in the introduction, let V be a rank 7 • and E • .The first flag, F • , allows us to define Schubert loci in the flag bundle Fl γ (V ) as in §2.4.The second flag, E • , determines a section s of Fl γ (V ) → X, and we define degeneracy loci as scheme-theoretic inverse imag s under s:
D w = s −1 Ω w ⊂ X.
When X is Cohen-Macaulay and D w has expected codimension (equal to the length of w; see §A.3), we have
[D w ] = s * [Ω w ] = P w (x; y) (2.6)
in H * X, where x i = −c 1 (E i /E i−1 ) and y i = c 1 (F i /F i−1 ).More generally, this polynomial defines a class supported on D w , even without assumptions on the singularities of X or the genericity of the flags F • and E • ; see [Fu1] or [Fu-Pr, App.A] for the intersection-theoretic det (z 1 , z 2 ) → diag(z 1 , z 2 , z 1 z −1 2 , 1, z −1 1 z 2 , z −1 2 , z −1 1
). Write 1 and t 2 for the corresponding weights.Then T prese ves γ and acts on
F l γ (V ). The total equivariant Chern class of V is c T (V ) = (1 − t 2 1 )(1 − t 2 2 )(1 − (t 1 − t 2 ) 2 ), so we have H * T (F l γ (V ), Z[ 1 2 ]) = Z[ 1 2 ][x 1 , x 2 , t 1 , t 2 ]/(r 2 , 2 , (x 1 − x 2 ) 2 ) − e i (t 2 1 , t 2 2 , (t 1 − t 2 ) 2 ).
A presentation with Z coefficients can be deduced from Theorem 5.4; see Remark 5.5.

Theorem 2.4 yields an equivariant Giambelli formula:

[Ω w ] T = P w (x; t) in H * T F l γ .In fact, this formula holds with integer coefficients: the Schubert classes form a basis for H * T (F l γ , Z) over Z[t 1 , t 2 ], so in particular there is no torsion, and
H * T (F l γ , Z) includes in H * T (F l γ , Z[ 1 2 .5, assume V has trivial determinant and γ has values in the trivial bundle, so the structure group is G = G 2 .The data of two γ-isotropic flags in V gives a map to the classifying space BB × BG BB, where B ⊂ G is a Borel subgroup, and there are universal degeneracy loci Ω w in this space.On the other hand, there is an isomorphism
BB × BG BB ∼ = EB × B (G/B), carrying Ω w to EB × B Ω w . Since H * T (F l γ ) = H * (EB × B (G/B)), and [Ω w ] T = [EB × B Ω w ], a Giambelli formula for [Ω w ]
T is equivalent to a degeneracy locus formula for this situation.One may then use equivariant localization to verify a given formula; this is essentially the approach taken in [Gr2].2.7.Other types.It is reasonable to hope for a similar degeneracy locus story in some of the remaining exceptional types.Groups of type F 4 and E 6 are closely relate

to Albert algebras, and bundle
ersions of these algebras have been defined and studied over some one-dimensional bases [Pu].Concrete realizations of the flag varieties have been given for types F 4 [La-Ma], E 6 [Il -Ma], and E 7 [Ga].Part of the challenge is to produce a complete flag from one of these realizations, and this seems to become more dif icult as the dimension of the minimal irreducible representation increases with respect to the rank.


Octonions and compatible forms

Any description of G 2 geometry is bound to be related to octonion algebras, since the simple group of type G 2 may be realized as the automorphism group of an octonion algebra; see Proposition 3.2 below.For an entertaining and wideranging tour of the octonions (also known as the Cayley numbers or octaves), see [Ba].

The basic inear-algebraic data can be defined as in §2, without reference to octonions, but the octonionic description is equivalent and sometimes more concrete.In this section, we collect the basic facts about octonions that we will use, and establish their relationship with the notion of compatible forms introduced in §2.1.Most of the statements hold over an arbitrary field, but we will continue to assume k is algebr ically closed of characteristic not 2.

While studying holonomy groups of Riemannian manifolds, Bryant proved several related facts about octonions and representations of (real forms of) G 2 .In parti

lar, he gives a
ay of producing a compatible bilinear form associated to a given trilinear form; we will use a version of this construction for fo ms on vector bundles.See [Br] or [Ha] for a discussion of the role of G 2 in differential geometry.

As far as I am aware, the results in § §3.2-3.3 have not appeared in the literature in this f rm, although related ideas about trilinear forms on a 7-dimensional vec ng to [Sp-Ve, §1] for proofs of any non-obviou assertions.

Definition 3.1.A composition algebra is a k-vector space C with a nondegenerate quadratic norm N : C → k and an algebra structure m :
C ⊗ C → C, with identity e, such that N (uv) = N (u)N (v).
Denote by β ′ the symmetric bilinear form associated to N , defined by
β ′ (u, v) = N (u + v) − N (u) − N (v). (Notice that β ′ (u, u) = 2N (u).) Since N (u) = N (eu) = N (e)N (u) for all u ∈ C, it follows that N (e) = 1 and β ′ (e, e) = 2.
The possible dimensions for C are 1, 2, 4, and 8.A composition algebra of dimension 4 is called a quaternion algebra, nd one of dimension 8 is an octonion algebra; octonion algebras are neither associative nor commutative.If For u, v ∈ V , we have β ′ (u, v)e = N (u + v)e − N (u) − N (v)e = −uv − vu. (3.3) Although C may not be associative, we always have u(uv) = (uu)v = N (u)v and (uv)v = u(vv) = N (v)u for any u, v ∈ C. Also, for u, v, w ∈ C we have β ′ (uv, w) = β ement u ∈ C is a zerodivisor if there is a nonzero v such that uv = 0. We have 0 = u(uv) = (uu)v = N (u)v, so u is structure on C corresponds to a trilinear form
γ ′ : C ⊗ C ⊗ C → k, using β ′ to identify C with C * . Specifically, we note by γ. (This follows fr C, with C = k ⊕ V , is characterized by m(u, v) = − 1 2 β(u, v)e + γ(u, v, •) † for u, v ∈ V ; (3.7) m(u, e) = m(e, u) = u for u ∈ V ; (3.8)
m(e, e) = e.(3.9)Conversely, given a trilinear form γ ∈ 3 V * and a nondegenerate bilinear form β ∈ Sym 2 V * , extend β(u, v)e, − 1 2 β(u, v)e) + 1 2 β ′ (γ(u, v, •) † , γ(u, v, •) † ) = 1 4 β(u, v)β(u, v) + 1 2 γ(u, v, γ(u, v, •) † ),
and
N (u)N (v) = 1 4 β(u, u)β(v, v).
Remark 3.4.Similar characterizations of octonionic multiplication have been given, usually in t γ and β are compatible forms on V , defining a composition a gebra structure on
C = k ⊕ V . Then L ⊂ V is γ-isotropic iff uv = 0 in C for all u, v ∈ L.
In particular y u ∈ L, choose a nonzero v ∈ u ⊥ ∩ L. Since L is γ-isotropic, γ(u, v, •) † = 0, so u and v are zerodivisors uv = − 1 2 β(u, v) e + γ(u, v, •) † = 0.
Therefore N (u) = N (v) = 0, so N and β are zero on . By (3.7), this also implies uv = 0 for all u, v ∈ L.

Finally, it will be convenient to use certain bases for C and V .We need a well-known lemma: In fact, given any a ∈ V = e ⊥ ith N (a) = 1, we can choose b and c so that a, b, c is an orthonormal basic triple; similarly, if a and b are orthonormal vectors generating a quaternion subalgebra, we can find c so that a, b, c is an orthonormal basic triple.

If a, b, c are an orthonormal b sic triple, let {e 0 = e, e 1 , . . ., e 7 } be the corresponding basis (in the same order as in Lemma 3.6).This is a standard orthonormal basis for C. With respect to the basis {e 1 , . . ., e 7 } for the imaginary octonions V , we have β(e p , e q ) = 2 δ pq , and (Here e * p is the map e q → δ pq .
tions for defining the octonionic product in terms of a standard basis vary widely in the literature, though -Coxeter [Co,p. 562] calculates 480 possible variations!A choice of convention corresponds to a labelling and orientation of the Fano arrangement of 7 points and 7 lines; the one we use agrees with that of p. 363].(Coincidentally, our choice of γ very nearly agrees with the one used in [Br,§2]: there the signs of e * 347 and e * 356 are positive, and the common factor of 2 is absent.)

We will most often use a different basis.Define
f 1 = 1 2 (e 1 + i e 2 ) f 2 = 1 2 (e 5 + i e 6 ) f 3 = 1 2 (e 4 + i e 7 ) f 4 = i e 3 f 5 = − 1 2 (e 4 − i e 7 ) f 6 = − 1 2 (e 5 − i e 6 ) f 7 = − 1 2 (e 1 − i e 2 ),(3.11)
and call this the standard γ-isotropic basis for V .(Here i is a fixed square root of −1 in k.)With respect to this basis, the bilinear for e expression (3.13) to compute 2 , f 3 , •) † . Since γ(f 2 , f 3 , f j ) = −δ 7,j = β(f 1 , f j ), we see γ(f 2 , f 3 , •) † = f 1 ther c metric bilinear form defined as in (2.2) (for char(k) = 3), by composing First suppose γ is nondegenerate.Since U is a GL(V )-orbit in 3 V * , we may choose a basis {f j } so that γ has the expression (3.13).Computing in this basis, a p , f 8−q ) = −δ pq for p, q = 4, and β γ (f 4 , f 4 ) = −2.Indeed, we have
(u, v) → − 1 3 γ(u, •, •) ∧ γ(v, •, •) ∧ γ with an isomorphism 7 V * ∼ = k.γ(f 1 , •, •) ∧ γ(f 7 , •, •) ∧ γ = (f * 47 − f * 56 ) ∧ (f * 14 − f * 23 ) ∧ γ = 3f * 1234567 .
The others are similar.In particular, with this choice of isomorphism 7 V * ∼ = k, γ and β γ are compatible forms.(For an arbitrary choice of isomorphism, β γ is a scalar multiple of a compatible form.)

To see this works in characteristic 3, one can avoid division by 3. Let V Z be a rank 7 free Z-module, fix a basis f 1 , , •, •) ∧ γ Z (f 4 , •, •) ∧ γ Z = 6 f *
1234567 , so one can define β γ over Z. (For nondegeneracy, one still needs char(k) = 2 here.)

For the converse, note that the terms in the compatibility relation (2.1) make sense for all γ in U ′ , since here γ(u, v, •) † is well-defined.We have seen that the relation holds on the dense open subset U ⊂ U ′ , so it must hold on all of U ′ .Therefore every γ in U ′ has a c mas prove Proposition 2.2:
Lemma 3.10. If u ∈ V is a nonzero isotropic vector, then E u = {v ∈ V | uv = 0} = {v ∈ V | γ(u, v, •) ≡ 0} is a three-dimensional β-isot opic subspace.
Proof.By definition, E u consists of zero-divisors, so it is β-isotropic by (3.5).Since β is nondegenerate on V , we know dim E u ≤ 3.

In fact, it is enough to observe that G = Aut(C) acts transitively on the set of isotropic vectors (up to scalar); this follows from Proposition A.5.Thus for any u, we can f nd g ∈ G such that g • u = λf 1 for some λ = 0.
uch that {u, v, w} is a basis.Then vw = λu for some nonzero λ ∈ k.
Proof. First note that vw = −wv, since −vw − wv = β(v, w)e = 0. If {u, v ′ , w ′ } is another basis, with v ′ = a 1 u + a 2 v + a 3 w and w ′ = b 1 u + b 2 v + b 3 w, then a 2 b 3 − a 3 b 2 = 0, so v ′ w ′ = (a 2 b 3 )vw + (a 3 b 2 )wv = (a 2 b 3 − a 3 b 2 )vw
is a nonzero multiple of vw.Now it suffices to check this for the standard γ-isotropic basis, and indeed, we c mputed
f 2 f 3 = f 1 in Example 3.8. Corollary 3.12. Let V = L 1 ⊕ • • • ⊕ L 7 be a splitting into one-dimensional subspaces such that L 1 is γ- = (k * ) 2 act on V via the matrix
diag(z 1 , z 2 , z 1 z −1 2 , 1, z −1 1 z 2 , z −1 2 , z −1 1 ) (in the f -basis).
Then T preserves the forms β an γ of (3.12) and (3.13).

The corresponding weights for this torus action are {t 1 , t 2 , t 1 − t 2 , 0, t 2 − t 1 , −t 2 e over X is a vector bundle C → X, equipped with a nondegenerate quadratic orm N : C → k X , a multiplication m : C ⊗ C → C, and an identity section e : k X → C, such that N respects composition.(Equivalently, for each x ∈ X, the fiber C(x) is a composition algebra ov r k.)

Since char(k) = 2, there is a corresponding nondegenerate bilinear form β ′ on C. We will also allow composition algebras whose norm takes values in a line bundle M ⊗2 ; here the multiplication is
C ⊗ C m − → C ⊗ M ,C ⊗ C m -C ⊗ M M ⊗4 . N ⊗ (N • e) N ⊗ N - The norm of e is the quadratic map M → M ⊗2 corresponding to M ⊗2 β ′ − → M ⊗2 . Replacing C with C = C ⊗ M * ,
one obtains a composition algebra whose norm takes values in the trivial bundle.

Many of the properties of composition algebras discussed above have straig tforward generalizations to bundles; we mention a few without giving proofs.

Using β ′ to identify C with C * ⊗ M ⊗2 , the multiplication map corresponds to a trilinear form γ ′ : C ⊗ C ⊗ C → M ⊗3 .The imaginary subbundle V is the orthogonal complement to e in C, so C = M ⊕ V .The bilinear form β ′ restricts to a nondegenerate form β on V , and 3 V → M ⊗3 .As before, the multiplication on M ⊕ V can be recovered from the forms β and γ on V , and there is an analogue of Proposition 3.3.

The analogues of Proposition 2.2 and Corollary 3.
2 can be proved using octonion bundles and reducing to the local case: Proposition 3.15.Let γ : 3 V → M ⊗3 and β : Sym 2 V → M ⊗2 be (locally) compatibl forms.Let F 1 ⊂ V be a γ-isotropic line bundle, and let ϕ : V → F * 1 ⊗ V * ⊗ M ⊗3 be the map defined by γ.Then the bundle
E F 1 = ker(ϕ)
has rank 3 and is β-isotropic.

Proposition 3.16.Let V be as in Proposition 3.15, and suppose there is a splitting
V = L 1 ⊕ • • • ⊕ L 7 into line bundles such that L 1 is γ-isotropic,andL 1 ⊕ L 2 ⊕ L 3 = E L 1 . Then the map V ⊗ V → V * ∼ = V ⊗ M induced by γ and β restricts to an isomorphism L 2 ⊗ L 3 ∼ − → L 1 ⊗ M .
Remark 3.17.Composition algebras may defined over an arbitrary base scheme X; in fact, as with Azumaya algebras, one is mainly interested in cases whe e X is defined over a non-algebraically closed field or a Dedekind ring.Petersson has classified such composition algebra bundles in the case where X is a cur .


Topology of G 2 flags

There are two "γ-isotropic Grassmannians" parametrizing γ-isotropic subspaces of dimensions 1 or 2, which we write as Q or G, respectively; thus F l γ embeds in Q×G.Since γ-isotropic vectors are just those v such that β(v, v) = 0, Q is the smooth 5-dimensional quadric hypersurface in P(V ).

Proposition 4.1.The γ-isotropic flag variety is a smooth, 6-dimensional projective variety.Moreover, both projections F l γ → Q

d F l γ → G are P 1bun
les.

Proof.The quadric Q comes with a tautological line bundle S 1 ⊂ V Q .By Proposition 2.2, the form γ also equips Q with a rank-3 bundle S 3 ⊂ V Q , with fiber S 3 ([u]) = E u , the space swept out by all γ-isotropic 2-spaces containing u. Thus S 1 ⊂ S 3 , and from the definitions we have F l γ (V ) = P(S 3 /S 1 ) → Q. (We use the convention that P(E) parametrizes lines in the vector bundle E.)

Similarly, if S 2 is the tautological bundle on G, e have F l γ (V ) = P(S 2 ) → G.This also shows that G is smooth of dimension 5.

Remark 4.2.The definition of F l γ (V ) can be reformulated as follows.Let F l = F l(1, 2; V ) be the two-step partial flag variety.The nondegenerate form γ is also a section of the trivial vector bundle 3 V * on F l.By restriction it gives a section of the rank 5 vector bundle 2 S * 2 ⊗ Q * 5 , where S 1 ⊂ S 2 ⊂ V is the tautological flag on F l and Q 5 = V /S 2 .Then F l γ ⊂ F l is defined by the vanishing of this section.

Remark 4.3.Projective y, F l γ parametrizes data (p ∈ ℓ), where ℓ is a γisotropic line in Q, and p ∈ ℓ is a point.Thus Proposition 2.2 says that the union of such ℓ through a fixed p is a P 2 in Q, and conversely, given such a P 2 one can recover p (as the intersection of any two γ-isotropic lines in the P 2 ).4.1.Fixed points.Let {f 1 , f 2 , . . ., f 7 } be the standard γ-isotropic basis for V , and let T = (k * ) 2 act as in Lemma 3.13, via the matr x diag(z 1 , z 2 , z 1 z −1 2 , 1, z −1 1 z 2 , z −1 2 , z −1 1 ).Write e(i j) for the two-step flag f i ⊂ f i , f j .

Proposition 4.4.This action of T defines an action on F l γ (V ), with 12 fixed points:

e(1 2), e(1 3), e(2 1), e(2 5), e(3 1), e(3 6), e(5 2), e(5 7), e(6 3), e(6 7), e(7 5), e(7 6).

Proof.Sin

T preserves β
it acts on Q, fixing the 6 points
[f 1 ], [f 2 ], [f 3 ], [f 5 ], [f 6 ], [f 7 ].
Since T preserves γ, it acts on F l γ , and the projection F l γ → Q is T -equivariant.The T -fixed points of F l γ lie in the fibers over the fixed points of Q.Since each of these 6 fibers is a P 1 with nontrivial T -action, there must be 2 • 6 = 12 fixed points.

To see the fixed points are as claimed, note that the bundle S 3 on Q is equivariant, nd the fibers S 3 (x) = E x at each of the fixed points are as f 1 , f 5 E f 3 = f 3 , f 1 , f 6 E f 5 = f 5 , f 2 , f 7 E f 6 = f 6 , f 3 , f 7 E f 7 = f 7 , f 5 , f 6 .
Indeed, one simply checks that in each triple, the (octonionic) product of the first vector with either the second or the third is zero.(Alternatively, one can ompute directly using the form (3.13).)Now the T -fixed lines in S 3 ([
f i ])/S 1 ([f i ]) are [f j ]
, where f j is the second or third vector in the triple .

In general, the T -fixed points of a flag variety are indexed by the corresponding Weyl group W , which for type G 2 is the dihedral group with 12 elements.We will write elements as w = w(1) w(2), for w(1) and w(2) such that e(w(1) n Proposition 4.4.We fix two simple reflections generating W , s = 2 1 and t = 1 3. See §A.3 for more details on the Weyl group and its embedding in S 7 .4.2.Schubert varieties.Fix a (complete) γ-isotrop c flag F • in V .Each Tfixed point is the center of a Schubert cell, which is defined by
X o w = {E • ∈ F l γ | dim(F p ∩ E q ) = r w (q, p) for 1 ≤ q ≤ 2, 1 ≤ p ≤ 7}
, where r w (q, p) = #{i ≤ q | w(i) ≤ p}, just as in the classical types.Also as in classical types, these can be parametrized by matrices, where E i is the span of the first i rows.For example, the big cell is
X o 7 6 = X a b c d e 1 Y Z S T f 1 0 ∼ = A

,
where lowercase variables are free
n by
X = −ae − bd − c 2 Y = −a − bf + cd − cef Z = −cf − d 2 + def S = c rs as imaginary octonions, the condition is that their product be zero.In fact, X, Y, Z are already determined by β-isotropicity.)Parametrizations of the other 11 cells ar rieties X w are the closures of the Schubert cells; equivalently,
X zations of cells, we see dim X w = ℓ(w).To get Schubert varieties with codimension ℓ(w), define Ω w = X w w 0 .These can also be described using the tautological quotient bundles:
Ω w = {x ∈ F l γ | rk(F p (x) → Q q (x)) ≤ r w (q, p)}.
Schubert varieties in Q and G are defined by the same conditions.(Note that w and w s define the same varieties in G, and w and w t define the exception of X 1 2 , all Schubert varieties in F l γ are inverse images of Schubert varieties in Q or G: Proposition 4.5.Let p : F l γ → Q and q : F l γ → G be the projections.Then X w = p −1 X w if w(1) > w(2).
The proof is immediate from the definitions.For instance, X tst = X 36 is a P 2 in Q: it parametrizes all 1-dimensional subspaces of a fixed isotropic 3-space.Its inverse image in F l γ is p −1 X tst = X tst = Ω sts .


Cohomology of flag bundles

5.1.Compatible forms on bundles.Let V be a rank 7 vector bundle o a variety X, equipped with a nondegenerate form γ : 3 V → L, and let B γ :
Sym 2 V → det V * ⊗ L ⊗3 be the Bryant form ( §2.1, §2.3). Assume there is a line bundle M such that det V * ⊗ L ⊗3 ∼ = M ⊗2 .
(5.1) (For example, this holds if V has a maximal B γ -isotropic subbundle F , for then we can take M = F ⊥ /F .There exist Zariski-locally trivial bundles V without this property, tho

h -see p. 293].)Lemma 5.1.I
this setup, L ∼ = M ⊗3 ⊗ T , for some line bundle T such that T ⊗3 is trivial.If L has a cube root, then T is trivial and
M ∼ = det V ⊗ (L * ) ⊗2 .
The proof is straightforward; see [An1,Lemma 3.2.1]for details.

From now on, we will assume V has a maximal B γ -isotropic subbundle F = F 3 ⊂ V .We also assume L has a cube root on X, so L ∼ = M ⊗3 .(By a theorem of Totaro, one can always assume this so long as 3-torsion is ignored in Chow groups (or cohomology); see [Fu2].In the case at hand, Lemma 5.1 gives a direct reason.)5.2.A splitting principle.For the next three subsections, we assume the line bundle M is trivial; this implies det V is e is the following:

Lemma 5.2.Assume V is equipped with a nondegenerate trilinear g, we can use β to identify L 8−i with L * i , and Proposition 3.16 implies
L 3 ∼ = L 1 ⊗ L * 2 . Thus V ∼ = L 1 ⊕ L 2 ⊕ (L 1 ⊗ L * 2 ) ⊕ k X ⊕ (L * 1 ⊗ L 2 ) ⊕ L * 2 F 2 ⊂ V , we have
V ∼ = F 2 ⊕ (F 1 ⊗ (F 2 /F 1 ) * ) ⊕ k X ⊕ (F * 1 ⊗ (F 2 /F 1 )) l base for V (with the assumed restrictions) is BGL 1 × BGL 1 .5.3.Chern classes.We continue to assume the line bundle M is trivial, and let (5.3)where y i = c 1 (L i ), and also that
F 1 ⊂ F 2 ⊂ F 3 ⊂ V be a γ-isotropic flag in V . It follows from (5.2) that c(V ) = (1 − y 2 1 )(1 − Q(V ) → X be the quadric bundle, with 1 ⊂ S 3 ⊂ V . Set x 1 = −c 1 (S 1 ) and α = [P(F 3 )] in H * Q(V ). The classes 1, x 1 , x 2 1 , α, x 1 α, x 2 1 α form a basis for H * Q(V ) over H * X; see Appendix B. Lemma 5.3. We have c 1 (S 3 ) = −2 x 1 , and c 2 (S 3 ) = 2 x 2 1 + c 2 (F 3 ) − 2 c 1 (F sifying space for this setup is BGL 1 × BGL 1 , and there is no torsion in its cohomology, it follows that the formula also holds with integer coefficients.5.4.Presentations.Using the fact that Fl γ (V ) is a P 1 -bundle over a qu

V ∼
M ⊗7 and the corresponding bilinear form has values in M ⊗2 .The splitting principle (Lemma 5.2) holds as stated for γ : 3 V → M ⊗3 .The compatible bilinear form β now identifies L 8−i with L * i ⊗ M ⊗2 , and we have
L 3 ∼ = ace has no torsion in cohomology; it follows that we may deduce integral formulas using rational coefficients.

As described in [Fu2], this situation reduces to the case where L is trivial.
Let V = V ⊗ M * , so γ : 3 → L determines a form γ : 3 V → k X . If V = L 1 ⊕ • • • ⊕ L 7 is a γ-isotropi ⊗ M * .Therefore x i = −c 1 ( S i / S i−1 ) = x i + v.The presentation for H * Fl γ (V ) is obtained from Proposition 5

efficients
n its Schubert expansion are nonnegative combinations of monomials in the positive roots.It is therefore natural to ask whether this is the equivariant class of a T -invariant subvariety of F l γ .In fact, it is the class of a T -equivariant embedding of SL 3 /B. 1 Remark 7.3.Graham's polynomial yields a simpler formula for the case where γ takes values in the trivial bundle, but det V = M is not necessarily trivial.(In this case, recall that M ⊗3 is trivial.)Making the substitutions x i → x i + v and y i → y i − v, with 3 v = 0, we obtain
[Ω w 0 ] = 1 54 (2 x 1 − x 2 − y 1 + 2 y 2 )(2 x 1 − x 2 − y 1 − y 2 )(x 1 − 2 x 2 + y 1 + y 2 ) ×(2 x 3 1 − 3 x 2 1 x 2 − 3 x 1 x 2 2 + 2 x 3 2 − 2 y 3 1 + 3 y 2 1 y 2 + 3 y 1 y 2 2 − 2 y 3 2 + v 3 ).
There is a more transparent choice of polynomial representative for [Ω w 0 ] ∈ H * F l γ (i.e., the case where the base is a point): he class of a point in the 5-dime omials P w for Schubert classes [Ω w ] using divided difference operators; see [An1,§4.2] for th polynomials using divided dif erence operators.In this respect, the problem of "G 2 Schubert polynomials" is worse than the situation for types B and C: they cannot even satisfy two of Fomin-Kirillov's conditions [Fo-Ki]. 2 Specifically, we have the following:

1 This embedding ions we consider are (3)] and a stronger version of [Fo-Ki, All the remaining facts are standard, and can be found in e.g.[Fu-Ha], [Hu1], [Hu2], [De].

A.1.General facts.Let G be a simple linear algebraic group, fix a maximal torus and Borel subgroup T ⊂ B ⊂ G, and let W = N (T )/T be the Weyl group.L t R, R + , and ∆ be the corresponding roots, positive roots, and simple roots, respectively.For α ∈ ∆, let s α ∈ W be the corresponding simple reflection, and also write s α ∈ N (T ) for a choice of lift; nothing in what follows will depend on the choice.For a subset S ⊂ ∆, let P S be the parabolic subgroup generated by B and {s α | α ∈ S}. (Such parabolic subgroups are called standard.)Write ı = ∆ {s i }, so P b ı is the maximal parabolic in which the ith simple root is omitted.

(For example, SL 5 /P b 2 ∼ = Gr(2, 5).) Write g, b, t, p, for the corresponding Lie algebras.

The length of an element w ∈ W is the least number ℓ = ℓ(w) such that w = s 1 • • • s ℓ (with s j = s α j for some α j ∈ ∆); such a minim l expression for w is called a reduced expression.Let w 0 be the (unique) longest element of W .The Bruhat order on W is defined by setting v ≤ w if there are reduced expressions
v = s β 1 • • • s β ℓ(v) and w = s α 1 • • • s α ℓ(w)
such that the β's are among the α's.

For each w ∈ W , there is a Schubert cell X o w = BwB/B in G/B, of dimension ℓ(w).The Schubert varieties X w are the closures of cells, and
X v ⊆ X w iff v ≤ w.
The irreducible representations of G are indexed by dominant weights; write V λ for the representation corresponding to the dominant weight λ.In characteristic 0, if p λ ∈ P(V λ ) is the point corresponding to a highest weight vector, then G • p λ is the unique closed orbit, and is identi ied with G/P S(λ) , where S(λ) is the set of simple roots orthogonal to λ with respect to a W -invariant inner product.In positive characteristic, G/P S(λ) can still be embedded in P(V ) for some representation with highest weight λ, but V need not be irreducible.(See [Hu2,§31] for these facts about representations in arbitrary characteristic.)A.2. Representation theory of G 2 .The root system of type G 2 has simple roots α 1 and α 2 (with α 2 the long root), and positive roots α 1 , α 2 , α 1 +α 2 , 2α 1 + α 2 , 3α 1 + α 2 , 3α 1 + 2α 2 .The lattice of abstract weigh s is the same as the root lattice (cf.[Hu2, §A.9]); it follows that up to isomorphism, here i only one simple group of type G 2 (over the algebraically closed field k).From now on, let G denote this group, and fix T ⊂ B ⊂ G corresponding to the root data.By Proposition 3.2, G ∼ = Aut(C), where C is the unique o tonion algebra over k.Let V = e ⊥ ⊆ C be the imaginary subspace.
he dominant Weyl chamber for this choice of positive roots is the cone spanned by α 4 and α 6 ; denote these fundamental weights by ω 1 and ω 2 , entation g has highest weight ω 2 .(This is irreducible if char(k) = 0, but not if char(k) = 3.)Over any field, one has g ⊆ 2 V .

Let γ be the alternating trilinear form on V ⊂ C induced by the multiplication, let {f 1 , . . ., f 7 } be the standard γ-isotropic basis (3.11).From the description of G as the automorphisms of C, it is clear that G preserves γ.In fact, the converse is almost true: Proposition A.1.Choose a basis {f 1 , . . ., f 7 } for V , and let γ ∈ 3 V * be given by as in (3.13).Let G(γ) ⊂ GL(V ) be the stabilizer of γ under the natural action, and let SG
γ = f * 147

f * 246 + f * 345 − f *
156 − f * 237 ,(γ) = G(γ) ∩ SL(V ). Then SG(γ) is simple of type G 2 , and G(γ) = µ 3 × SG(γ). Moreover, the orbit GL(V ) • γ is open in 3 V * .
For k = C, this is well known; see [Br,§2] or [Fu-Ha, §22].For arbitrary fields, see [An1,Propositions 6.1.4 and A.2.2], and compare [As,(3.4)].

The proof of this proposition also shows the following:

Corollary A.2. Let V , γ, and SG(γ) be as in Proposition A.1.Then SG(γ) acts irreducibly on V .

Note that w 0 ∈ W acts on the weight lattice by multiplication by −1.This implies that every irreducible representa ion of G is isomorphic to its dual.Using Schur's lemma, there is a unique (up to scalar) G-invariant bilinear form on each irreducible representation [Hu2,§31.6].In particular, we have the following: Proposition A.3.Let V be a 7-dimensional vector space, with nondegenerate trilinear form γ : 3 V → k.Then γ determines a compatible form β uniquely up to scaling by a cube root of unity.

Remark A.4.In characteristic 0, the description of G 2 (or g 2 ) as the stabilizer of a generic alternating trilinear form is due to Engel, who also found an invariant symmetric bilinear form.For a history of some of the early constructions of G 2 , see [Ag].

A.3.The Weyl group.The Weyl group of type G 2 is the dihedral group with 12 elements.Let α 1 and α 2 be the simple roots, and let s = s α 1 and t = s α 2 be the corresponding simple reflections generating W = W (G 2 ).Thus W has a presentation s, t | s 2 = t 2 = (st) 6 = 1 .With the exception of w 0 , each element of W (G 2 ) has a unique reduced expression.The Hasse diagram for Bruhat order is as follows: w 0 = 7 6 (tststs = ststst) (ststs) 7 5 6 7 (tstst) (tsts) 6 3 5 7 (stst) (sts) 5 2 3 6 ( st)
(ts) 3 1 2 5 (st) (s) 2 1 1 3 (t) id = 1 2
The indexing w = w(1) w( 2), for 1 ≤ w(1), w(2) ≤ 7, arises as follows.There is an embedding W S 7 , given by s → τ 12 τ 35 τ 67 and t → τ 23 τ 56 , where τ ij is the permutation transposing i and j. (This also factors through W (B 3 ).)Thus each w is identified with a permutation w(1) w(2) • • • w(7), and in fact, the full permutation is determined by w(1) w( 2).

This inclusion of Weyl groups corresponds to the inclusion G 2 ֒→ SL 7 determined by the basis {f 1 , . . ., f 7 } for V = V ω 1 and the trilinear form γ of (3.13), together with the inclusion of tori
(z 1 , z 2 ) → (z 1 , z 2 , z 1 z −1 2 , 1, z −1 1 z 2 , z −1 2 , z −1 1 )
. Thus a natural way to extend w ∈ W to a full permutation is as follows.Given w(1) w(2), et w(3) be the number such that E f w(1) = f w(1) , f w(2) , f w(3) as in §4.1.Then define w(4), . . ., w(7) by requiring w(i) + w(8 − i) = 8.For example, 6 3 extends to 6 3 7 4 1 5 2. Note that (w • w 0 )(i) = 8 − w(i).

All this can be summarized in the following diagram:

1 2 3 4 5 6 7

1 2 2 1 1 3 2 5 3 1 5 2 3 6 5 7 6 3 7 5 6 7 7 6 s t A.4. Homogeneous spaces.We can now identify the homogeneous spaces for G 2 .We take G = Aut(C) for an octonion algebra C, as above, and let β and γ be the corresponding compatible forms on the imaginary subspace V ⊂ C. Proof.The homogeneous spaces G/P b 1 and G/P b 2 are the closed orbits in P(V ) and P(g), respectively.Since G preserves β, G/P b 1 must be contained in the quadric hypersurface Q ⊂ P(V ), but dim G/P b 1 = 5, so it is all of Q.

To see G/P b 2 = G, note that G/P b 2 ⊂ P(g) ⊂ P( 2 V ), so G/P b 2 ⊂ Gr(2, 7).Since G preserves γ, we must have G/P b 2 ⊆ G; thus it will suffice to show G is irreducible and 5-dimensional.For this, consider
F l γ = {(p, ℓ) | p ∈ ℓ} ⊂ Q × G,
and notice that the first projection identifies F l γ with the P 1 -b ndle P(S 3 /S 1 ) → Q, so F l γ is smooth and irreducible of dimension 6.On the other hand, the second projection is obviously a P 1 -bundle.

Finally, since
F l γ is a 6-dimensional G-invariant subvariety of G/P b 1 × G/P b 2 , it follows that F l γ = G/B./B = (B ′ i(w)B ′ /B ′ ) ∩ (G/B).
More generally, let P ⊂ G and P ′ ⊂ G ′ be parabolic subgroups such that P = P ′ ∩ G. Then the same conclusion holds for G/P ֒→ G ′ /P ′ , that is, BwP/P = (B ′ i(w)P ′ /P ′ ) ∩ (G/P ) for all w ∈ W .

A.5.The Borel map and divided differences.Let M ⊂ t * be the weight latt ced by the Chern class map c 1 : M → H 2 (G/B), where M ⊂ t * is the weight lattice.More precisely, this map is efined as follows.Identify M with the character group of B, and associate to χ ∈ M the line bundl L χ = G × B C. Then c 1 (χ) is defined to be c 1 (L χ ).(See [BGG, De].)In fact, c 1 is an isomorphism, and this induces an action of W in the evident way: for w ∈ W and
x = c 1 (χ) ∈ H 2 (G/B), define w • x = c 1 (w • χ).
The Borel map becomes surjective after extending scalars to Q, and defines an isomorphism
H * (G/B, Q) ∼ = Sym * M Q /I, where I = (Sym * M Q ) W
+ is the ideal of positive-degree Weyl group invariants.For a simple root α, define the divided difference operator ∂ α on H * (G/B) by
∂ (f ) = f − s α • f α . (A.1)
These act on Schubert classes as follows [De]:
∂ α [Ω w ] =
[Ω w sα ] when ℓ(w s α ) < ℓ(w); 0 when ℓ(w s α ) > ℓ(w).(A.2)

In particular, [Ω sα ] can be identified with the weight at the intersection of the hyperplanes orthogonal t α and the (affine) hyperplane bisecting α.

In the case of G 2 flags, we know [Ω s ] = x 1 and [Ω t ] = x 1 + x 2 .Looking at the root diagram, then, we see x 1 = α 4 and x 2 = α 3 .Therefore
α 1 = x 1 − x 2 , α 2 = −x 1 + 2x 2 , and s • x 1 = x 2 , s • x 2 = x 1 , t • x 1 = x 1 , t • x 2 = x 1 − x 2 .
With these substitutions, the operators of (A.1) agree with those defined in §6 ((2.4) and (2.5)).


Appendix B Integral Chow rings of quadric bundles

In this appendix, we consider schemes over an arbitrary field k, and use the language of Chow ri gs rather than cohomology.We prove the following fact about odd-rank quadric bundles: Theorem B.1.Let V be a vector bundle of rank 2n + 1 on a scheme X, and suppose V is equipped with a nondegenerate quadratic form.Assume there is a maximal (rank n) isotropic subbundle F ⊂ V .Let Q A similar presentation for even-rank quadrics was first given by Edidin and Theorem 7]; in fact, the seco d of the two relations is the same as theirs.Our purpose here is to correct a small error in the statement of the second half of their theorem (which concerned odd-rank quadrics).

Before giving the proof, we recall two basic formulas for Chern classes.Let L be a line bundle.For a vector bundle E of rank n, we have (cf.[Fu4,Ex. 3.2.2])
c n (E ⊗ L) = n i=0 c i (E) c 1 (L) n−i . (B.3) Also, if 0 → L → E → E ′ → 0
is an exact sequence of vector bundles, then inverting the Whitney formula gives
c k (E ′ ) = c k (E) − c k−1 (E) c 1 (L) + • • • + (−1) k c 1 (L) k . (B.4)
Proof.The classes h, h 2 , . . ., h n−1 , f, f h, . . ., f h n−1 form a ba is of A * Q as an A * X-module, since they form a basis when restricted to a fiber.It is easy to see that these elements also form a basis of the ring A * X[h, f ]/I.Therefore it suffices to establish that the relations generating I hold in A * Q.

Let i : Q ֒→ P(V ) be the inclusion of the quadric in the projective bundle.By [Fu4,Ex. 3.2.17To prove the first relation, expand h n in the given basis:
h n = a 0 f + a 1 h n−1 + • • • + a n , (B.6)
with a k ∈ A k X.Our goal is to show a 0 = 2, and a k = (−1) k+1 c k (F ) for k > 0.

That a = 2 can be seen by restricting to a fiber: the Chow ring of an odd-dimensional quadric in projective space is given by Z[h, f ]/(h n − 2f, f 2 ).

Multiplying (B.6) by h and expanding in the basis, we have h n+1 = 2 h f + 2 a 1 f (a 2 + a 2 1 ) h n−1 + • • • + (a n + a 1 a n−1 ) h + a 1 a n .On the other hand, if we rearrange and expand (B.5), we obtain
h n+1 = 2 h f − 2 c 1 f − (c 2 + c 1 a 1 )h n−1 − • • • − (c n + c 1 a n−1 ) h − (c ients, we have 2 a 1 = −2 c 1 ;
a k = −c k − a k−1 (a 1 + c 1 ) (2 ≤ k ≤ n); a 1 a n = −c n+1 − c 1 a 1 .
From the first of these equations, we see
a 1 + c 1 = τ,
for some τ ∈ A 1 X such that 2 τ = 0. (Note that τ = 0 only if c n+1 (V /F ) = 0, which need not be true in general.)The remaining equations give
a k = −c k + c k−1 τ − c k−2 τ 2 + • • • − (−1) k τ k (1 ≤ k ≤ n), (B.7)
and −c n+1 = a n τ .(Of course, t e signs on powers of τ make no difference, but we will include them as a visual aid.)

We claim τ = c 1 (F ⊥ /F ).This can be proved in the universal case.Specifying the maximal isotropic subbundle F ⊂ V reduces the structure group from O(2n+1) to a parabolic subgroup whose Levi factor is GL n ×Z/2Z, so dle over) BGL n × BZ/2Z.Now A * (BGL n × BZ/2Z) ∼ = Z[c 1 , . . ., c n , t]/(2t), so there is only one 2-torsion class of degree 1, namely t.

Since See [To] for the meaning and computation of this Chow ring.To ensure V is pulled back from Totaro's algebraic model for BG, one may have to replace X by an affine bundle or Chow envelope, as in [Gr1,p. 486].)

Using the exact sequence 0 → F ⊥ /F → V /F → V /F ⊥ → 0 and Formula (B.4), Equation (B.7) implies
a k = −c k (V /F ⊥ ).
Since V /F ⊥ ∼ = F ∨ , we obtain a k = (−1) k+1 c k (F ), as desired.

The second relation is proved by the argument given in [Ed-Gr].Let j : P(F ) ֒→ Q be the inclusion, and let N P(F )/Q be the normal bundle.By the selfintersection formula, j * c n (N (F )/Q ) = f 2 .On the other hand, using N Q/P(V ) = O(2) and N P(F )/P(V ) = V /F ⊗ O(1), and tensoring with O(−1), we have 0 → O(1) → V /Q ⊗ O(−1) → 0 on P(F ); thus N P(F )/Q = ((V /F )/O(1)) ⊗

′ (u, u) if and only if γ and β are compatible, in
he sense of Definition 2.1.




Lemma 3.6 ([Sp-Ve, (1.6.3)]).There are elements a, b, c ∈ C such that e, a, b, ab, c, ac, bc, (ab)c forms an orthogonal basis for C.Such a triple is cal ed a basic triple for C.




and the identity is M e − → C.Here a little care is required in the definition.The composition C ⊗ M id⊗e − −− → C ⊗ C m − → C ⊗ M should be the identity, and the other composition (m • (e ⊗ id)) should be the canonical isomorphism.The compatibility between m and N is encoded in the commutativity of the following diagram:




(1)].description agrees with the Lie-theoretic one.Propositions A.1 and A.3 are the basic representation-theoretic facts relating G 2 to compatible forms; their proofs are given in

An1, A
pendix A].Proposition A.5 identifies the γ-isotropi flag variety F l γ with the homogeneous space G 2 /B.




From the root data, one sees dim G = 14, dim B = 8, dim P b 1 = dim P b 2 = 9, and dim T = 2. Thus dim G/B = 6 and dim G/P b 1 = dim G/P b 2 = 5.Proposition A.5.Let F l γ , Q, and G be as in §4.Then Q ∼ = G/P b 1 , G = G/P b 2 , and F l γ ∼ = G/B.




Remark A.6.A similar description of G/P b 2 , among others, can be found in[La-Ma].Proposition A.7.Let i : G ֒→ G ′ be an inclusion of semisimple algebraic groups, and let B ⊂ G and B ′ ⊂ G ′ be Borel subgroups with i(B) ⊂ B ′ .Also denote by i the induced inclusions of flag varieties G/B ֒→ G ′ /B ′ and Weyl groups W ֒→ W ′ .Then for each w ∈ W , the Schubert cells are related by BwB


p−→

X be the quadric bundle of isotropic lines in V , l t h ∈ A * Q be the hyperplane class (restricted from H = c 1 (O(1)) ∈ A * P(V )), and let f = [P(F )] ∈ A s generated by the two relations2f = h n − c 1 (F ) h n−1 + • • • + (−1) n c n (F ), (B.1) f 2 = (c n (V /F ) + c n−2 (V /F ) h 2 + • • • ) f. (B.2)(Here h and f have degrees 1 and n, respectively.)




], we havei * f = [P(F )] = n+1 i=0 c i H n+1−i in A * P(V ), where c i = c i (V /F ).(Following the common abuse of notation, we have writt n c i for p * c i .)On the other hand,Q ⊂ P(V ) is cut out by a section of O P(V ) (2), so [Q] = 2 H in A * P(V ).Therefore i * i * = 2 h f , and we have 2 h f = h n+1 + c 1 h n + • • • + c n+1 .(B.5) (Up to this point, we are repeating the argument of [Ed-Gr].)




Then γ is nondegene ate if and only if β γ is nondegenerate.(Infact, β γ is also defined if char(k) = 3, and the same conclusion holds.)Proof.Let U ⊂ 3 V * be the set of nondegenerate forms, and let U ′ ⊂ 3 V * be the set of forms γ such that β γ is nondegenerate; we want to show U = U ′ .(By Proposition A.1, U is open and dense.) with projections p s i : Z s → Fl γ and p t i : Z t → Fl γ .The proofs of the following two lemmas are the same as in cl ssical types; see [An1, §4.1] for details.
Lemma 6.2. As maps
If char(k) = 2, the representation V = e ⊥ ⊂ C contains an invariant subspace spanned by e. In this case, the irreducible representation Vω 1 = V /(k • e) is 6-dimensional [Sp-Ve, §2.3].
This work was partially supported by an RTG fellowship, NSF Grant 0502170.By making the substitutions x i → x i + v and y i → y i − v, we obtain form las for the more general case, where γ has values in M ⊗3 for arbitrary M .Theorem 6.4.Let γ : 3 V → M ⊗3 be a nondegenerate form, with a γisotropic flag F • ⊂ V .Let v = c 1 (M ).Let ∂ s be defined as above, and let ∂ t be given bywhere P w = ∂ w 0 w −1 P w 0 , andVariationsAny formula for the class of a degeneracy locus depends on a choice of representative modulo the ideal defining the cohomolo y ring; here we discuss some alternative formulas.In type A, the Schubert polynomials of Lascoux and Schützenberger are generally accepted as the best polynomial representatives for Schubert classes and degeneracy loci: they have many remarkable geometric and combinato ial (and aesthetic) properties.In other classical types, several choices have been proposed -see, e.g., [Bi-Ha, La-Pr, K -Ta, Fo-Ki, Fu2]but Fomin and Kirillov[Fo-Ki]gave examples showing tha no choice can satisfy all the properties possessed by the type A polynomials.From this point of view, an investigation of alternative G 2 formulas could shed some light on the problem for classical types, by imposing some limitations on what one might hope to find for general Lie types.Proposition 7.