chow rings of complex algebraic groupskaji/papers/chow-ohp1.pdf · g: simply connected simple...

Chow rings of Complex Algebraic Groups

Shizuo Kajijoint with

Masaki Nakagawa

Symplectic Geometry Seminarat University of Toronto

Nov. 26, 2007

Chow rings of algebraic groups

Outline

10 days ago, Duan and Zhao posted a preprint(arXiv:math.AT/0711.2541v1) which announces a series of resultssimilar to those I will discuss today.

Introduction

Cohomology of flag variety

Borel presentationSchubert presentation

Connection between the two presentations

Divided difference operator

Computations and Main Theorems

Future Work


Notations

G : simply connected simple complex algebraic group(a complexification of a simple 1-connected compact Lie group)

B: Borel subgroup of G

G/B is a projective variety called the flag variety

H∗(G/B;Z): ordinary integral cohomology of G/B

A∗(G ): Chow ring of G


Main goal

General Goal

Determine A∗(G) for all simply connected simple complex algebraic groups

Classification Theorem tells that G is one of the following types:SLn,Spinn,Spn,G2,F4,E6,E7,E8

Chevalley and Grothendieck considered the problem in the 1950’s.

They gave a formula to compute it from H∗(G/B;Z).Consequently, A∗(G) was determined to be trivial for G = SLn, Spn.

A∗(G )⊗ Z/p was determined by Kac(1985) for all G .

A∗(G ) for G = Spinn,G2,F4 were determined by R.Marlin(1974).

He resorted to Schubert calculus to determine H∗(G/B;Z).His method seems to be hopeless for other exceptional types.(Note: Nakagawa also checked the result of Marlin by the samemethod we use here).

Our Goal Today

Determine A∗(G ) for G = E6,E7,E8.


Chow ring

A∗(X ): the Chow ring of a non-singular variety X

A∗(X ) =⊕

i≥0 Ai (X )

Ai (X ) (= Adim X−i (X )) is a group of the rational equivalence classesof algebraic cycles of codimension i .(an algebraic cycle is a linear sum of possibly singular subvarieties)

intersection product Ai (X )⊗ Aj(X ) → Ai+j(X )

cycle map cl : A∗(X ) → H2∗(X ;Z): ring homomorphism(Note: taking the Poincare dual of the fundamental class of a cycle)


Basic Facts

Theorem (Grothendieck(1958))

the cycle map is an isomorphism of rings:

A∗(G/B)'−→ H2∗(G/B;Z).

the pullback of the projection p : G → G/B induces a surjectionp∗ : A∗(G/B) → A∗(G ),where the kernel is an ideal generated by A1(G/B).

Corollary

A∗(G ) ∼= H∗(G/B;Z)/(H2(G/B;Z))

Note: Since H∗(G/B;Q) is generated by degree 2 elements,A∗(G )⊗Q = Q for all G .For G = SLn,Spn, H∗(G/B;Z) is also generated by degree 2 elements,and so A∗(G ) = Z.


Strategy

A presentation for H∗(G/B;Z) was given by Borel.

It is called Borel presentation, which is a quotient of a polynomialring divided by some ideal.Ring structure is clear, but generators have little geometric meaning.

H∗(G/B;Z) has another module basis consisting of by Schubertclasses.

Schubert classes come from subvarieties.But we don’t know structure constants, so it is difficult to useGrothendieck’s Theorem.

Hence, what we will do are:

1 Compute A∗(G ) purely algebraically from Borel presentation.

2 Find Schubert varieties representing the generators.

Main tool

We use the divided difference operator given by Demazure andBerstein-Gelfand-Gelfand.


Borel presentation

K : maximal compact subgroup of G (=real compact form of G )

T : maximal compact torus of K (=(S1)l)

BT : classifying space of T (=(CP∞)l)

W : Weyl group of K (=N(T )/T )

{ωi}1≤i≤l : fundamental weights and H∗(BT ;Z) = Z[ω1, . . . , ωl ]

Inclusion K ↪→ G induces a diffeomorphism K/T ∼= G/B.

the classifying map K/Tι−→ BT of the T -bundle T → K → K/T

induces the characteristic map ι∗ : H∗(BT ;Z) −→ H∗(K/T ;Z)

Theorem (Borel(1953))

1 ι∗ : H∗(BT ;Q) −→ H∗(K/T ;Q) is surjective.

2 The kernel is (H+(BT ;Q)W ) an ideal generated by theW -invariants of positive degrees.


Borel presentation

Toda(1975) extended Borel’s work to give H∗(K/T ;Z) by a quotientring of a polynomial ring.Based on Toda’s method, H∗(K/T ;Z) were explictly determined:

· H∗(SU(n)/T ;Z) · · ·Borel (1953)

· H∗(Spin(n)/T ;Z) · · ·Toda-Watanabe (1974)

· H∗(Sp(n)/T ;Z) · · ·Borel (1953)

· H∗(G2/T ;Z) · · ·Bott-Samelson (1955)

· H∗(F4/T ;Z) · · ·Toda-Watanabe (1974)

· H∗(E6/T ;Z) · · ·Toda-Watanabe (1974)

· H∗(E7/T ;Z) · · ·Nakagawa (2001)

· H∗(E8/T ;Z) · · ·Nakagawa (2007, preprint)


Schubert presentation

The Bruhat decomposition of G

G =∐

w∈W

BwB

gives a cell decomposition

G/B =∐

w∈W

BwB/B.

l(w): length of w ∈ W , w0 ∈ W : the longest element

X ◦w = BwB/B ∼= Cl(w): Schubert cell

Xw = closure of X ◦w : Schubert variety

Zw = {Poincare dual of the fundamental class [Xw0w ]} ∈H2l(w)(G/B;Z): Schubert class

{Zw}w∈W forms an additive basis for H∗(G/B;Z).


Comparison of the two presentations

Hence we have two descriptions forH∗(K/T ;Z) = H∗(G/B;Z) = A∗(G/B)

Borel presentation Schubert presentation

elements polynomials Schubert classes

geometry no algebraic cycles

ring structure easy hard

Demazure and BGG’s divided difference operator bridges those twopresentations.



K : maximal compact subgroup of G (=real compact form of G )

T : maximal compact torus of K (=(S1)l)

Π = {αi}1≤i≤l : simple roots

{ωi}1≤i≤l : fundamental weights ( (2αj

(αj ,αj ), ωi ) = δij )

si : simple reflection corresponding to αi

( si (e) = e − ( 2αi

(αi ,αi ), e)αi )

W : Weyl group of G (= a finite group generated by {si}1≤i≤l)

Definition (B-G-G(1973), Demazure(1973))

1 For αi ∈ Π, ∆i : H∗(BT ;Z) → H∗−2(BT ;Z)

∆i (f ) =f − si (f )

αi, f ∈ H∗(BT ;Z) = Z[ω1, . . . , ωl ]

2 For w ∈ W , w = si1si2 · · · sik : a reduced decomposition∆w = ∆i1 ◦∆i2 ◦ · · · ◦∆ik



Theorem (B-G-G(1973), Demazure(1973))

∆w : H∗(BT ;Z) → H∗−2l(w)(BT ;Z) is well-defined.

A map c : H2k(BT ;Z) → H2k(K/T ;Z) defined by

c(f ) =∑

l(w)=k

∆w (f )Zw (Note:∆w (f ) ∈ Z)

is identical to the characteristic map ι∗. cf

How to calculate ?

∆α(ωβ) = δαβ

∆α(fg) = ∆α(f )g + sα(f )∆α(g)


Translation

Note: H∗(K/T ;Z) = H∗(G/B;Z) has no torsion (Bott(1958)).Borel H∗(K/T ;Q) Schubert

||H∗(K/T ;Z) ⊂ H∗(BT ;Q)/(H+(BT ;Q)W ) ⊃ H∗(K/T ;Z)

|| || ||Z[ω1, . . .]/(ρ1, . . .) ⊂ Q[ω1, . . . , ωl ]/(φ1, . . .) ⊃

Lw∈W Z{Zw}

(representative) ↘ ↑ ↗ (characteristic)Q[ω1, . . . , ωl ] = H∗(BT ;Q)


Convenient presentation of H∗(BT ;Z)

K = El (l = 6, 7, 8)

{ωi}1≤i≤l : fundamental weights

H∗(BT ;Z) = Z[ω1, ω2, . . . , ωl ]

We take another set of generators for H∗(BT ;Z):tl = ωl

ti = si+1(ti+1) =

{ωi − ωi+1 (4 ≤ i ≤ l − 1)

ωi−1 + ωi − ωi+1 (i = 2, 3)

t1 = s1(t2) = −ω1 + ω2

t = ω2


Convenient presentation of H∗(BT ;Z)

Let ci = i-th elementary symmetric function in t1, . . . , tl (1 ≤ i ≤ l)

H∗(BT ;Z) = Z[ω1, ω2, . . . , ωl ]

= Z[t1, t2, . . . , tl , t]/(c1 − 3t).

si (i 6= 2) act on {ti}1≤i≤l as permutations and trivially on t.

The action of s2 on {ti}1≤i≤l , and t is given by

s2(ti ) =

{t − t1 − t2 − t3 + ti (1 ≤ i ≤ 3)

ti (4 ≤ i ≤ l)

s2(t) = 2t − t1 − t2 − t3.

Thus, for f ∈ Z[t, c2, . . . , cl ], ∆i f = 0 if i 6= 2.

We sometimes call {t1, t2, . . . , tl , t} Toda-Watanabe’s magical basis.


Borel presentation for H∗(E6/T ;Z)

Theorem (Toda-Watanabe(1974))

H∗(E6/T ;Z) =Z[t1, t2, . . . , t6, t, γ3, γ4]

(ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ8, ρ9, ρ12)(|γi | = 2i)

ρ1 =c1 − 3t ρ2 = c2 − 4t2

ρ3 =c3 − 2γ3 ρ4 = c4 + 2t4 − 3γ4

ρ5 =c5 − 3tγ4 + 2t2γ3 ρ6 = γ32 + 2c6 − 3t2γ4 + t6

ρ8 =3γ42 − 6tγ3γ4 − 9t2c6 + 15t4γ4 − 6t5γ3 − t8

ρ9 =2c6γ3 − 3t3c6

ρ12 =3c26 − 2γ4

3 + 6tγ3γ42 + 3t2c6γ4 + 5t3c6γ3 − 15t4γ4

2 − 10t6c6

+ 19t8γ4 − 6t9γ3 − 2t12


Correspondence

Using the characteristic map, we can translate the generators{t1, t2, . . . , t6, t, γ3, γ4} in Borel presentation into Schubert classes.

Borel Schubert Borel Schubertt1 −Z1 + Z2 t6 Z6

t2 Z1 + Z2 − Z3 t Z2

t3 Z2 + Z3 − Z4 γ3 Z342 + 2Z542

t4 Z4 − Z5 γ4 Z1342 + 2Z3542 + Z6542

t5 Z5 − Z6

Furthermore, we wish to take a single Schubert class for each generator.In this E6 case, for example, we can take the following classes:

Z342 = −γ3 + 2t3

Z1342 = γ4 − 2tγ3 + 2t4


Finding a set of ring generators

**How to determine which Schubert classes can be chosen as generators?This question can be formulated as follows.

Definition

R: ring

R◦: non-invertible elements of R

decomposable ideal: R◦ · R◦

x ∈ R is indecomposable when x 6= 0 ∈ R/R◦ · R◦

In our setting when R = H∗(K/T ;Z):

There is at most one ring generator in each degree H∗>2(K/T ;Z).

If we find an indecomposable Zw ∈ H∗(K/T ;Z), then we take it asa generator.

Related question

Which Schubert classes are indecomposable ?


A(E6)

By Grothendieck’s Theorem,

A∗(G ) = A∗(G/B)/(A1(G/B))

= H2∗(G/B;Z)/(H2(G/B;Z))

= H2∗(K/T ;Z)/(H2(K/T ;Z)),

(Note: H2∗−1(K/T ;Z) = 0)where

H∗(K/T ;Z) = Z[t1, . . . , tl , t, γi1 , . . .]/(ρj1 , . . .)

H2(K/T ;Z) = Z{t1, . . . , tl , t}

Therefore to obtain A∗(G ) from H∗(K/T ;Z), we simply putt = 0, ti = 0, (1 ≤ i ≤ l) in Borel presentation.

H∗(E6/T ;Z)/(t1, . . . , t6, t) = Z[γ3, γ4]/(2γ3, 3γ4, γ23 , γ

34)

(using the correspondence) = Z[Z542,Z6542]/(2Z542, 3Z6542,Z2542,Z

36542)


Main Theorems

p : G → G/B: projectionSince a Schubert class Zw corresponds to Schubert variety Xw0w , we have

Theorem (K-Nakagawa)

A(E6) = Z[X3,X4]/(2X3, 3X4,X23 ,X 3

4 )

X3 = p∗(Xw0s5s4s2) = B(w0s5s4s2)B ⊂ GX4 = p∗(Xw0s6s5s4s2) = B(w0s6s5s4s2)B ⊂ G


H∗(E7/T ;Z)

Similarly, from the Borel presentation of H∗(E7/T ;Z), we have

H∗(E7/T ;Z)/(t1, . . . , t7, t)

= Z[γ3, γ4, γ5, γ9]/(2γ3, 3γ4, 2γ5, γ23 , 2γ9, γ

25 , γ

34 , γ

29)

= Z[Z542,Z6542,Z76542,Z654376542]/2Z542, 3Z6542, 2Z76542,Z2542, 2Z654376542,

Z 276542,Z

36542,Z

2654376542


A(E7)

Theorem (K-Nakagawa)

A(E7) = Z[X3,X4,X5,X9]

/(2X3, 3X4, 2X5,X23 , 2X9,X

25 ,X 3

4 ,X 29 )

X3 = p∗(Xw0s5s4s2) = B(w0s5s4s2)B ⊂ GX4 = p∗(Xw0s6s5s4s2) = B(w0s6s5s4s2)B ⊂ GX5 = p∗(Xw0s7s6s5s4s2) = B(w0s7s6s5s4s2)B ⊂ GX9 = p∗(Xw0s6s5s4s3s7s6s5s4s2) = B(w0s6s5s4s3s7s6s5s4s2)B ⊂ G


A(E8)

Proposition (K-Nakagawa)

A(E8) = Z[X3,X4,X5,X6,X9,X10,X15]/2X3, 3X4, 2X5, 5X6, 2X9,X

25 − 3X10,

X 34 , 2X15,X

29 , 3X 2

10,X83 ,

X 215 + X 3

10 + 2X 56

Xi = p∗(γi ) (i = 3, 4, 5, 6, 9, 10, 15)

Note: here Xi may not be the pull-back of a single Schubert variety but alinear combination of them.


Future Work

Determine which Schubert classes belong to the decomposable ideal.(equivalently, find indecomposable Schubert classes)

Find a presentation of a given Schubert class Zw as a polynomial ina fixed set of ring generators.(Schubert polynomial of type G2,G4,El(l = 6, 7, 8))

Replace B with any parabolic subgroup P in the above problems.(Note: there is a ring monomorphism H∗(G/P;Z) ↪→ H∗(G/B;Z)described in terms of Schubert presentation)

Note: Once a set of simple roots is fixed, a parabolic group Pcorresponds to a subset ΠP of simple roots. G/P has a Schubert basis{Zw}w∈S indexed by the left coset of W (G ) by W (P), where W (P) isgenerated by simple reflections corresponding to the complement of ΠP .S can be injected into W (G ) as {w ∈ W (G )|l(wsi ) > l(w),∀si ∈ ΠC

P},and this induces a ring monomorphism H∗(G/P;Z) ↪→ H∗(G/B;Z).


Example (finding indecomposables)

H∗(F4/B;Z) has generators only in degrees 2, 6, and 8.

H2(F4/B;Z) is spanned by Zw , where W = [1], [2], [3], [4], thelength one elements in the Weyl group. Of course they areindecomposable.

Out of 16(= dim H6(F4/B;Z)), the indecomposables are:

W = [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 2], [3, 2, 1], [3, 2, 3]

Out of 25(= dim H8(F4/B;Z)), the indecomposables are:

W =[1, 2, 3, 4], [1, 2, 3, 2], [1, 2, 4, 3], [1, 3, 2, 3], [1, 3, 2, 4], [1, 4, 3, 2]

[2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 2, 1], [2, 3, 2, 4], [2, 4, 3, 2]

[3, 2, 1, 3], [3, 2, 1, 4], [3, 2, 3, 4]

[4, 3, 2, 1], [4, 3, 2, 3]

Note: there are more than one way to express an element of Weyl groupby the products of the simple reflections.


Thank you for listening

and

for your patience with my ”exceptional” English


chow rings of complex algebraic groupskaji/papers/chow-ohp1.pdf · g: simply connected simple...

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