Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and...

Schubert calculus and cohomology of Lie groups

Haibao Duan, Institute of Mathematics, CAS

Toric Topology 2011 in Osaka§November 28, 2011

Haibao Duan, Institute of Mathematics, CAS Schubert Calculus

Outline of the talk

Problem (E. Cartan, 1929) Given a compact, connected Lie

group G§determine its cohomology H∗(G ; F) with coeffcients in

either F = R,Fp, or Z.

1929-1949: Results of Brauer, Pontryagin, Hopf, Samleson

and Yan for the case of F = R;

1950-1978: Results of Borel, Araki, Toda, Mimura, Kono for

the case of F = Fp;

Recent works of Duan and Zhao for the case of F = Z

Outline of the talk

the case of F = Fp;

Outline of the talk

the case of F = Fp;

Outline of the talk

1 Preliminaries

2 Earlier results

3 Schubert calculus

4 New results (Duan, Zhao)

1. Preliminaries 1: Cartan’s classification on Lie groups

Any compact, connected and finite dimensional Lie group G

admits the canonical form:

(G1 × · · · × Gk × T r )/K ,

in which

1 each Gi is one of the next 1-connected simple Lie groupsµ

SU(n),Sp(n),Spin(n),G2,F4,E6,E7,E8;

2 T r = S1 × · · · × S1 is the r−dimensional torus;

3 K is a finite subgroup of the center of G1 × · · · × Gk × T r .

Therefore§we can assume in this talk that

”G is one of the 1-connected simple Lie groups listed above.”

(G1 × · · · × Gk × T r )/K ,

in which

(G1 × · · · × Gk × T r )/K ,

in which

(G1 × · · · × Gk × T r )/K ,

in which

(G1 × · · · × Gk × T r )/K ,

in which

(G1 × · · · × Gk × T r )/K ,

in which

1. Preliminaries 2: Algebras and rings

An algebra is a vector space V with a product V⊗

V → V .

A ring is an abelian group A with a product A⊗

A → A.

===Example 1>>> For a given a manifold M

the cohomology H∗(M; F) with field coefficients F = Fp or Ris an algebra;

the integral cohomology H∗(M; Z) is a ring.

V → V .

A → A.

V → V .

A → A.

V → V .

A → A.

===Example 2>>> Let x1, · · · , xn be a set of n graded elements, and

let F = R,Fp, or Z.

1 The graded exterior algebra (or ring) over F:

ΛF[x1, · · · , xn] (i.e. xixj = −xjxi )

2 The graded polynomial algebra (or ring) over F:

F[x1, · · · , xn];

3 The graded truncated polynomial algebra (or ring) over F :

F[x1, · · · , xn]/ 〈f1, · · · , fk〉where fr ∈ F[x1, · · · , xn], and where 〈f1, · · · , fk〉 is the ideal

generated by the polynomials f1, · · · , fk .

let F = R,Fp, or Z.

F[x1, · · · , xn];

let F = R,Fp, or Z.

F[x1, · · · , xn];

let F = R,Fp, or Z.

F[x1, · · · , xn];

let F = R,Fp, or Z.

F[x1, · · · , xn];

let F = R,Fp, or Z.

F[x1, · · · , xn];

F[x1, · · · , xn]/ 〈f1, · · · , fk〉

where fr ∈ F[x1, · · · , xn], and where 〈f1, · · · , fk〉 is the ideal

let F = R,Fp, or Z.

F[x1, · · · , xn];

1. Preliminaries 3: Hopf Algebra

A Hopf (co-)algebra is an algebra V⊗

V → V with a co-product

β : V → V⊗

===Example 3>>> Let G be a Lie group with multiplication

β : G × G → G . The induced algebra map

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

furnishes the cohomology H∗(G ; F) with the structure of a Hopf

(co-)algebra.

In contrast, the integral cohomology H∗(G ; Z) is in general

not a Hopf ring, because of

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

Ext(· · · , · · · )We will refer this structure as the

”near Hopf ring structure on H∗(G ; Z))”

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

β : V → V⊗

β : G × G → G .

The induced algebra map

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

not a Hopf ring,

because of

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

Ext(· · · , · · · )

We will refer this structure as the

β : V → V⊗

β∗ : H∗(G ; F) → H∗(G ; F)⊗

H∗(G ; F), F = R,Fp

(co-)algebra.

β∗ : H∗(G ; Z) → H∗(G ; Z)⊗

H∗(G ; Z)⊕

”near Hopf ring structure on H∗(G ; Z))”Haibao Duan, Institute of Mathematics, CAS Schubert Calculus

2. Earlier works 1. F = R (1925-1949)

Up to 1935, Brauer and Pontryagin computed the algebra

H∗(G ; R) for the cases G = SU(n),SO(n),Sp(n):

1 H∗(SO(2n + 1); R) = ∧R(y3, y7, · · ·, y4n−1)

2 H∗(U(n); R) = ∧R(y1, y3 · ··, y2n−1)

3 H∗(Sp(n); R) = ∧R(y3, y5 · ··, y4n−1)

2. Earlier works 1. F = R (1925-1949)

1 H∗(SO(2n + 1); R) = ∧R(y3, y7, · · ·, y4n−1)

2 H∗(U(n); R) = ∧R(y1, y3 · ··, y2n−1)

3 H∗(Sp(n); R) = ∧R(y3, y5 · ··, y4n−1)

2. Earlier works 1. F = R (1925-1949)

1 H∗(SO(2n + 1); R) = ∧R(y3, y7, · · ·, y4n−1)

2 H∗(U(n); R) = ∧R(y1, y3 · ··, y2n−1)

3 H∗(Sp(n); R) = ∧R(y3, y5 · ··, y4n−1)

2. Earlier works 1. F = R (1925-1949)

1 H∗(SO(2n + 1); R) = ∧R(y3, y7, · · ·, y4n−1)

2 H∗(U(n); R) = ∧R(y1, y3 · ··, y2n−1)

3 H∗(Sp(n); R) = ∧R(y3, y5 · ··, y4n−1)

2. Earlier works 1. F = R (1925-1949)

1 H∗(SO(2n + 1); R) = ∧R(y3, y7, · · ·, y4n−1)

2 H∗(U(n); R) = ∧R(y1, y3 · ··, y2n−1)

3 H∗(Sp(n); R) = ∧R(y3, y5 · ··, y4n−1)

2. Earlier works 1. F = R (1929-1949)

The idea of Hopf and Samleson£1941¤µGiven a graded algebra

A = ⊕k≥0Ak over R with a co-product β : A → A

⊗A (i.e. an

Hopf algebra over reals),

what does A looks like as an algebra?

Introduce the subset of ”the primative elements” in the algebra A

P(A) = {a ∈ A | β(a) = a⊗ 1⊕ 1⊗ a}.Since it is a real vector space, we can take vector space basis

x1, · · ·, xn; y1, · · ·, ym

for P(A) with deg(xi ) = even and deg(yi ) = odd.

===Classification Theorem I, Hopf, Samleson, 1941>>>

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

P(A) = {a ∈ A | β(a) = a⊗ 1⊕ 1⊗ a}.

Since it is a real vector space, we can take vector space basis

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

P(A) = {a ∈ A | β(a) = a⊗ 1⊕ 1⊗ a}.Since it is a real vector space,

we can take vector space basis

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

⊗A (i.e. an

x1, · · ·, xn; y1, · · ·, ym

A = R[x1, · · ·, xn]⊗ ∧R(y1, · · ·, ym).

2. Earlier works 1. F = R (1929-1949)

===Corollary 1>>> If G is a compact Lie group§then

H∗(G ; R) = ∧R(y1, · · ·, ym)

with deg(yi ) = odd.

===Corollary 2, Yan, 1949>>> Let G be an exceptional Lie group.

1 H∗(G2; R) = ∧R(y3, y11)

2 H∗(F4; R) = ∧R(y3, y11, y15, y23)

3 H∗(E6; R) = ∧R(y3, y9, y11, y15, y17, y23)

4 H∗(E7; R) = ∧R(y3, y11, y15, y19, y23, y27, y35)

5 H∗(E8; R) = ∧R(y3, y15, y23, y27, y35, y39, y47, y59)

2. Earlier works 1. F = R (1929-1949)

===Corollary 1>>> If G is a compact Lie group§then

H∗(G ; R) = ∧R(y1, · · ·, ym)

with deg(yi ) = odd.

===Corollary 2, Yan, 1949>>> Let G be an exceptional Lie group.

1 H∗(G2; R) = ∧R(y3, y11)

2 H∗(F4; R) = ∧R(y3, y11, y15, y23)

3 H∗(E6; R) = ∧R(y3, y9, y11, y15, y17, y23)

4 H∗(E7; R) = ∧R(y3, y11, y15, y19, y23, y27, y35)

5 H∗(E8; R) = ∧R(y3, y15, y23, y27, y35, y39, y47, y59)

2. Earlier works 2. F = Fp (1950-1978)

The idea of Borel (1950)µ

Given a graded algebra A = ⊕k≥0Ak

over a finite field Fp with a co-product β : A → A⊗

A (i.e. an

Hopf algebra over Fp),

===Classification Theorem II, Borel, 1952>>> If A be a finitely

generated co-algebra over the finite field Fp, then

A = B(x1)⊗ · · · ⊗ B(xn)

where each B(xi ) is one of the ”monogenic Hopf algebra over Fp”:

B(xi ) deg(xi ) odd deg(xi ) even

p 6= 2 ΛFp(xi ) Fp(xi )/(xpr

p = 2 F2(xi )/(x2r

i ) F2(xi )/(x2r

The idea of Borel (1950)µGiven a graded algebra A = ⊕k≥0Ak

A (i.e. an

A = B(x1)⊗ · · · ⊗ B(xn)

p = 2 F2(xi )/(x2r

i ) F2(xi )/(x2r

A (i.e. an

A = B(x1)⊗ · · · ⊗ B(xn)

p = 2 F2(xi )/(x2r

i ) F2(xi )/(x2r

A (i.e. an

A = B(x1)⊗ · · · ⊗ B(xn)

p = 2 F2(xi )/(x2r

i ) F2(xi )/(x2r

Borel (1953) computed H∗(G2; F2),H∗(F4; F2);

Araki (1960) computed H∗(F4; F3);

Toda, Kono, Mimura, Shimada (1973,75,76) obtained

H∗(Ei ; F2), i = 6, 7, 8;

Kono, Mimura (1975, 1977) obtained H∗(Ei ; F3), i = 6, 7, 8;

Kono (1977) obtained H∗(E8; F5)

H∗(Ei ; F2), i = 6, 7, 8;

The cohomology H∗(E8; Fp) of E8 is given by

if p = 2µ

F2[α3,α5,α9,α15]

〈α163 ,α8

5,α49,α

415〉

⊗ ΛF2(α17, α23, α27, α29)

if p = 3µ

F3[x8, x20]/⟨x38 , x

⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);

If p = 5µ

F5[x12]/⟨x512

⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)

if p = 2µ

F2[α3,α5,α9,α15]

〈α163 ,α8

5,α49,α

415〉

⊗ ΛF2(α17, α23, α27, α29)

if p = 3µ

F3[x8, x20]/⟨x38 , x

⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);

If p = 5µ

F5[x12]/⟨x512

⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)

if p = 2µ

F2[α3,α5,α9,α15]

〈α163 ,α8

5,α49,α

415〉

⊗ ΛF2(α17, α23, α27, α29)

if p = 3µ

F3[x8, x20]/⟨x38 , x

⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);

If p = 5µ

F5[x12]/⟨x512

⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)

if p = 2µ

F2[α3,α5,α9,α15]

〈α163 ,α8

5,α49,α

415〉

⊗ ΛF2(α17, α23, α27, α29)

if p = 3µ

F3[x8, x20]/⟨x38 , x

⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);

If p = 5µ

F5[x12]/⟨x512

⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)

if p = 2µ

F2[α3,α5,α9,α15]

〈α163 ,α8

5,α49,α

415〉

⊗ ΛF2(α17, α23, α27, α29)

if p = 3µ

F3[x8, x20]/⟨x38 , x

⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);

If p = 5µ

F5[x12]/⟨x512

⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)

2. Earlier works: Summation

There are two problems arising from the previous worksµ

1 Problem 1 (V. Kac (1985); James Lin (1987)): Is there a

single procedure to determine the Hopf algebra H∗(G ; F) for

all G and F = Fp?

2 Problem 2. Determine the (near Hopf) ring H∗(G ; Z) for the

most difficult and subtle cases of G = G2,F4,E6,E7,E8

===Remark>>> For the classical Lie groups

G = U(n),Sp(n),Spin(n)§the near Hopf rings H∗(G ; Z) have

been determined by Borel (1952) and Pitties (1991).

all G and F = Fp?

3. Schubert Calculus

Take a maximal torus T in G and

let ω1, · · · , ωn ⊂ H2(G/T ; F)

be a set of fundamental dominant weights of G .

In the Leray–Serre spectral sequence {E ∗,∗r (G ; F), dr} of the

fibration T → Gπ→ G/T one has

1 E ∗,∗2 (G ; F) = H∗(G/T )⊗ ΛF(t1, · · · , tn)2 the differential d2 : E ∗,∗2 (G ; F) → E ∗,∗2 (G ; F) is given by

d2(x ⊗ tk) = xωk ⊗ 1, x ∈ H∗(G/T ; F), 1 ≤ k ≤ n

Leray (1951) showed that E ∗,∗3 (G ; R) = H∗(G ; R).

Serre (1964) proved that E ∗,∗3 (G ; Zp) = H∗(G ; Zp).

Kac (1984) and Marlin (1991) conjectured that

E ∗,∗3 (G ; Z) = H∗(G ; Z). (This conjecture has been confirmed by

Duan and Zhao in this work).

Take a maximal torus T in G and let ω1, · · · , ωn ⊂ H2(G/T ; F)

d2(x ⊗ tk) = xωk ⊗ 1, x ∈ H∗(G/T ; F), 1 ≤ k ≤ n

1 E ∗,∗2 (G ; F) = H∗(G/T )⊗ ΛF(t1, · · · , tn)

2 the differential d2 : E ∗,∗2 (G ; F) → E ∗,∗2 (G ; F) is given by

d2(x ⊗ tk) = xωk ⊗ 1, x ∈ H∗(G/T ; F), 1 ≤ k ≤ n

E ∗,∗3 (G ; Z) = H∗(G ; Z).

(This conjecture has been confirmed by

d2(x ⊗ tk) = xωk ⊗ 1, x ∈ H∗(G/T ; F), 1 ≤ k ≤ n

Duan and Zhao in this work).Haibao Duan, Institute of Mathematics, CAS Schubert Calculus

Open Question (A. Weil, Foundations of algebraic geometry,

1962):

”The classical Schubert calculus amounts to the

determination of cohomology rings of flag manifolds”

1 Chevalley (1958): The classical/Schubert classes0on G/T

is an additive basis of the cohomology H∗(G/T ;Z )

2 Duan (2005): Determined the ”Multiplicative rule of Schubert

classes”

1962):

classes”

1962):

classes”

3. Schubert calculus

Lemma (Duan and Zhao, 2006) For any Lie group G there exist

a set {y1, · · · , ym} of Schubert classes on G/T with deg yi > 2

that the ring H∗(G/T ; Z) has the presentation

Z[ω1, · · · , ωn; y1, · · · , ym]/ 〈ei , fj , gj〉1≤i≤k;1≤j≤m

1 for each 1 ≤ i ≤ k, ei ∈ 〈ω1, · · · , ωn〉

2 for each 1 ≤ j ≤ m, the pair (fj , gj) of polynomials is related

to the Schubert class yj in the fashion

fj = pjyj + αj ; gj = ykj

j + βj

with pj ∈ {2, 3, 5} and αj , βj ∈ 〈ω1, · · · , ωn〉

a set {y1, · · · , ym} of Schubert classes on G/T with deg yi > 2 so

1 for each 1 ≤ i ≤ k, ei ∈ 〈ω1, · · · , ωn〉

j + βj

a set {y1, · · · , ym} of Schubert classes on G/T with deg yi > 2 so

1 for each 1 ≤ i ≤ k, ei ∈ 〈ω1, · · · , ωn〉

j + βj

4. New results

Starting from the above Lemma, we can actually construct the ring

H∗(G ; Z) instead of computing it.

In view of the fibration π : G → G/T , the set {y1, · · · , ym} of

Schubert classes on G/T specified in the Lemma gives rise to the

integral classes

xdeg yi:= π∗(yi ) ∈ H∗(G ; Z), 1 ≤ i ≤ m.

Granted with composition

〈ω1, · · · , ωn〉ι→ E 2k,1

3 (G ; F)κ→ H2k+1(G ; F)

the polynomials ei , αj , βj ∈ 〈ω1, · · · , ωn〉 yield the integral classes

%k := κ ◦ ι(ei ) ∈ H∗(G ; Z), k = deg ei − 1

%k := κ ◦ ι(pjβj − ykj−1j αj) ∈ H∗(G ; Z), k = deg βj − 1

· · ·CI := · · ·

4. New results

integral classes

xdeg yi:= π∗(yi ) ∈ H∗(G ; Z), 1 ≤ i ≤ m.

〈ω1, · · · , ωn〉ι→ E 2k,1

3 (G ; F)κ→ H2k+1(G ; F)

%k := κ ◦ ι(ei ) ∈ H∗(G ; Z), k = deg ei − 1

· · ·CI := · · ·

4. New results

integral classes

xdeg yi:= π∗(yi ) ∈ H∗(G ; Z), 1 ≤ i ≤ m.

〈ω1, · · · , ωn〉ι→ E 2k,1

3 (G ; F)κ→ H2k+1(G ; F)

%k := κ ◦ ι(ei ) ∈ H∗(G ; Z), k = deg ei − 1

· · ·CI := · · ·

4. New results

integral classes

xdeg yi:= π∗(yi ) ∈ H∗(G ; Z), 1 ≤ i ≤ m.

〈ω1, · · · , ωn〉ι→ E 2k,1

3 (G ; F)κ→ H2k+1(G ; F)

%k := κ ◦ ι(ei ) ∈ H∗(G ; Z), k = deg ei − 1

· · ·

CI := · · ·

4. New results

integral classes

xdeg yi:= π∗(yi ) ∈ H∗(G ; Z), 1 ≤ i ≤ m.

〈ω1, · · · , ωn〉ι→ E 2k,1

3 (G ; F)κ→ H2k+1(G ; F)

%k := κ ◦ ι(ei ) ∈ H∗(G ; Z), k = deg ei − 1

· · ·CI := · · ·

4. New results

Theorem 1. With respect to the ring presentation

H∗(G2) = ∆Z(%3)⊗ ΛZ(%11)⊕ τ2(G2),

τ2(G2) = F2[x6]+/

⟨x26

⟩⊗∆F2(%3)

and where

%23 = x6, x6%11 = 0

the reduced co–product ψ is given by

{%3, x6} ⊂ P(G2),

ψ(%11) = δ2(ζ5 ⊗ ζ5).

4. New results

H∗(F4) = ∆Z(%3)⊗ ΛZ(%11, %15, %23)⊕ τ2(F4)⊕ τ3(F4)

τ2(F4) = F2[x6]+/

⟨x26

⟩⊗∆F2(%3)⊗ ΛF2(%15, %23)

τ3(F4) = F3[x8]+/

⟨x38

⟩⊗ ΛF3(%3, %11, %15)

%23 = x6, x6%11 = 0, x8%23 = 0,

{%3, x6, x8} ⊂ P(F4)

ψ(%11) = δ2(ζ5 ⊗ ζ5) + x8 ⊗ %3

ψ(%15) = −δ3(ζ7 ⊗ ζ7),

ψ(%23) = δ3(ζ7 ⊗ ζ7x8 − ζ7x8 ⊗ ζ7).

4. New results

H∗(E6) = ∆Z(%3)⊗ΛZ(%9, %11, %15, %17, %23)⊕ τ2(E6)⊕ τ3(E6)

τ2(E6) = F2[x6]+/

⟨x26

⟩⊗∆F2(%3)⊗ ΛF2(%9, %15, %17, %23),

τ3(E6) = F3[x8]+/

⟨x38

⟩⊗ ΛF3(%3, %9, %11, %15, %17)

and where

%23 = x6, x6%11 = 0, x8%23 = 0,

{%3, %9, %17, x6, x8} ⊂ P(E6);

ψ(%11) = δ2(ζ5 ⊗ ζ5) + x8 ⊗ %3;

ψ(%15) = x6 ⊗ %9 − δ3(ζ7 ⊗ ζ7);

ψ(%23) = x6 ⊗ %17 + δ3(ζ7x8 ⊗ ζ7 − ζ7 ⊗ ζ7x8).

4. New results

Theorem 4. The ring H∗(E7) has the presentation

∆Z(%3)⊗ ΛZ(%11, %15, %19, %23, %27, %35)⊕ τ2(E7)⊕ τ3(E7)

τ2(E7) =F2[x6, x10, x18, CI ]

+⟨x26 , x

210, x

218,DJ ,RK ,SI ,J ,Ht,L

⟩⊗∆F2(%3)⊗ΛF2(%15, %23, %27)

with t ∈ e(E7, 2) = {3, 5, 9}, I , J, L ⊆ e(E7, 2), |I | , |J| ≥ 2,

τ3(E7) =F3[x8]

+⟨x38

⟩ ⊗ ΛF3(%3, %11, %15, %19, %27, %35)

and where

%23 = x6, x8%23 = 0

4. New results

{%3, x6, x8, x10, x18} ⊂ P(E7);

ψ(%11) = δ2(ζ5 ⊗ ζ5) + x8 ⊗ %3;

ψ(%15) = δ2(ζ9 ⊗ ζ5) + δ3(ζ7 ⊗ ζ7);

ψ(%19) = δ2(ζ9 ⊗ ζ9);

ψ(%23) = δ2(ζ17 ⊗ ζ5) + δ3(ζ7x8 ⊗ ζ7 − ζ7 ⊗ ζ7x8);

ψ(%27) = δ2(ζ17 ⊗ ζ9)− δ3(ζ7 ⊗ ζ19);

ψ(%35) = δ2(ζ17 ⊗ ζ17) + x8 ⊗ %27 − %27 ⊗ x8 + x8 ⊗ x8%19;

ψ2(ζ2i−1) = 0, i ∈ e(E7, 2).

4. New results

Theorem 5. The ring H∗(E8) has the presentation

∆Z(%3, %15, %23)⊗ ΛZ(%27, %35, %39, %47, %59) ⊕p=2,3,5

τp(E8)

τ2 = F2[x6,x10,x18,x30,CI ]+

〈x86 ,x4

10,x218,x

230,DJ ,RK ,SI ,J ,Ht,L〉 ⊗∆F2(%3, %15, %23)⊗ΛF2(%27)

with t ∈ e(E8, 2) = {3, 5, 9, 15}, K , I , J, L ⊆ e(E8, 2), |I | , |J| ≥ 2,

|K | ≥ 3;

τ3 =F3[x8,x20,C{4,10}]

+⟨x38 ,x3

20,x28 x2

20C{4,10},C2{4,10}

⟩ ⊗ ΛF3(%3, %15, %27, %35, %39, %47);

τ5 = F5[x12]+

〈x512〉

⊗ ΛF5(%3, %15, %23, %27, %35, %39, %47),

4. New results

and where

%23 = x6, %

215 = x30, %

223 = x6

x2s%3s−1 = 0, for s = 4, 5

x8%59 = x220C{4,10}, x20%23 = x2

8C{4,10},

x12%59 = 0,

{%3, x6, x8, x10, x12, x18, x20} ⊂ P(E8);

ψ(%15) = δ2(ζ9 ⊗ ζ5) + x26 ⊗ %3 − δ3(ζ7 ⊗ ζ7) + x12 ⊗ %3;

ψ(%23) = δ2(ζ17 ⊗ ζ5 +∑

s+t=2x s6ζ5 ⊗ x t

6ζ5) + x210 ⊗ %3

+δ3(x8ζ7 ⊗ ζ7 − ζ7 ⊗ ζ7x8)− δ5(ζ11 ⊗ ζ11);

4. New results

ψ(%27) = δ2(ζ17⊗ ζ9) + δ3(ζ19⊗ ζ7)− x12⊗ %15 + (x46 + 2x2

12)⊗ %3;

ψ(%35) = δ2(ζ17 ⊗ ζ17)− %27 ⊗ x8 + x8 ⊗ %27 + x20 ⊗ %15

+δ3(x8ζ19⊗ ζ7)+2x12⊗ρ23 + δ5(x12ζ11⊗ ζ11 +3ζ11⊗ ζ11x12);

ψ(%39) = δ2(∑

s+t=2x s10ζ9 ⊗ x r

10ζ9)− δ3(ζ19 ⊗ ζ19) + x12 ⊗ %27

+2x212 ⊗ %15 − x3

12 ⊗ %3;

ψ(%47) = δ2(∑

s+t=6x s6ζ5 ⊗ x r

6ζ5)− x20 ⊗ %27 + %39 ⊗ x8

+δ3(x20ζ19 ⊗ ζ7) + 2x12 ⊗ %35 + x212 ⊗ %23

+δ5(ζ11 ⊗ x212ζ11 +

∑s+t=2

x s12ζ11 ⊗ x r

12ζ11);

4. New results

ψ(%59) = δ2(x210ζ29⊗ ζ9 + x30ζ17⊗ ζ5x6 + x18ζ29⊗ ζ5x6 + x4

6 ζ29⊗ ζ5ζ29 ⊗ ζ29 + x2

10ζ17 ⊗ ζ9x26 + ζ17 ⊗ x2

6 ζ29 + x46 ζ17 ⊗ ζ5x

+x18ζ17 ⊗ ζ5x46 + x4

6x210 ⊗ ζ5ζ9 + x2

10 ⊗ ζ9ζ29 + x46 ⊗ ζ5ζ29)

δ3(∑

s+t=1(−x20)

sζ19⊗ x r20ζ19)+ 2δ5(

∑s+t=4

(−x12)sζ11⊗ x r

12ζ11);

and for (p, i) = (2, 3), (2, 5), (2, 9), (3, 4), (3, 10), (5, 6)

ψp(ζ2i−1) = 0;

ψ2(ζ29) = x210 ⊗ ζ9 + ζ17 ⊗ x2

6 + x46 ⊗ ζ5.

4. New results

Main idea in the computation:

It might appear difficult to compute with the cohomology classes

xi , %k and CI ,

but it is easier to calculate with the polynomials

ei , αj , βj ∈ 〈ω1, · · · , ωn〉.

Since the cohomology classes xi , %k and CI are constructed from

the polynomials ei , αj , βj in the Schubert classes, one can boil

down the calculation in the cohomology ring H∗(G ; Z) to the

computation with those polynomials.

4. New results

xi , %k and CI ,

ei , αj , βj ∈ 〈ω1, · · · , ωn〉.

4. New results

xi , %k and CI ,

ei , αj , βj ∈ 〈ω1, · · · , ωn〉.

4. New results

xi , %k and CI ,

ei , αj , βj ∈ 〈ω1, · · · , ωn〉.

the polynomials ei , αj , βj in the Schubert classes,

one can boil

4. New results

xi , %k and CI ,

ei , αj , βj ∈ 〈ω1, · · · , ωn〉.

Thanks!

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