schubert calculus and cohomology of lie groupsmasuda/toric/duan.pdf · schubert calculus and...
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![Page 1: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/1.jpg)
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
![Page 2: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/2.jpg)
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
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 3: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/3.jpg)
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
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 4: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/4.jpg)
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
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 5: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/5.jpg)
Outline of the talk
1 Preliminaries
2 Earlier results
3 Schubert calculus
4 New results (Duan, Zhao)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 6: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/6.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 7: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/7.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 8: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/8.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 9: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/9.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 10: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/10.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 11: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/11.jpg)
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.”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 12: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/12.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 13: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/13.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 14: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/14.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 15: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/15.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 16: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/16.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 17: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/17.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 18: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/18.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 19: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/19.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 20: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/20.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 21: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/21.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 22: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/22.jpg)
1. Preliminaries 2: Algebras and rings
===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 .
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 23: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/23.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 24: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/24.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 25: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/25.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 26: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/26.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 27: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/27.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 28: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/28.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 29: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/29.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 30: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/30.jpg)
1. Preliminaries 3: Hopf Algebra
A Hopf (co-)algebra is an algebra V⊗
V → V with a co-product
β : V → 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))”Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 31: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/31.jpg)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 32: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/32.jpg)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 33: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/33.jpg)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 34: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/34.jpg)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 35: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/35.jpg)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 36: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/36.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 37: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/37.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 38: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/38.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 39: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/39.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 40: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/40.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 41: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/41.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 42: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/42.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 43: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/43.jpg)
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.
Then
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 44: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/44.jpg)
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.
Then
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 45: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/45.jpg)
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),
what does A looks like as an algebra?
===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
i )
p = 2 F2(xi )/(x2r
i ) F2(xi )/(x2r
i )
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 46: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/46.jpg)
2. Earlier works 2. F = Fp (1950-1978)
The idea of Borel (1950)µGiven 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),
what does A looks like as an algebra?
===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
i )
p = 2 F2(xi )/(x2r
i ) F2(xi )/(x2r
i )
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 47: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/47.jpg)
2. Earlier works 2. F = Fp (1950-1978)
The idea of Borel (1950)µGiven 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),
what does A looks like as an algebra?
===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
i )
p = 2 F2(xi )/(x2r
i ) F2(xi )/(x2r
i )
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 48: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/48.jpg)
2. Earlier works 2. F = Fp (1950-1978)
The idea of Borel (1950)µGiven 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),
what does A looks like as an algebra?
===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
i )
p = 2 F2(xi )/(x2r
i ) F2(xi )/(x2r
i )
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 49: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/49.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 50: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/50.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 51: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/51.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 52: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/52.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 53: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/53.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 54: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/54.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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
320
⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);
If p = 5µ
F5[x12]/⟨x512
⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 55: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/55.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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
320
⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);
If p = 5µ
F5[x12]/⟨x512
⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 56: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/56.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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
320
⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);
If p = 5µ
F5[x12]/⟨x512
⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 57: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/57.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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
320
⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);
If p = 5µ
F5[x12]/⟨x512
⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 58: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/58.jpg)
2. Earlier works 2. F = Fp (1953-1978)
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
320
⟩⊗ ΛF3(ζ3, ζ7, ζ15, ζ19, ζ27, ζ35, ζ39, ζ47);
If p = 5µ
F5[x12]/⟨x512
⟩⊗ ΛF5(ζ3, ζ11, ζ15, ζ23, ζ27, ζ35, ζ39, ζ47)
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 59: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/59.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 60: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/60.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 61: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/61.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 62: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/62.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 63: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/63.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 64: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/64.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 65: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/65.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 66: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/66.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 67: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/67.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 68: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/68.jpg)
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 69: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/69.jpg)
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).Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 70: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/70.jpg)
3. 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”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 71: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/71.jpg)
3. 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”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 72: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/72.jpg)
3. 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”
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 73: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/73.jpg)
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
so
that the ring H∗(G/T ; Z) has the presentation
Z[ω1, · · · , ωn; y1, · · · , ym]/ 〈ei , fj , gj〉1≤i≤k;1≤j≤m
where
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〉
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 74: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/74.jpg)
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 so
that the ring H∗(G/T ; Z) has the presentation
Z[ω1, · · · , ωn; y1, · · · , ym]/ 〈ei , fj , gj〉1≤i≤k;1≤j≤m
where
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〉
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 75: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/75.jpg)
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 so
that the ring H∗(G/T ; Z) has the presentation
Z[ω1, · · · , ωn; y1, · · · , ym]/ 〈ei , fj , gj〉1≤i≤k;1≤j≤m
where
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〉
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 76: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/76.jpg)
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 := · · ·
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 77: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/77.jpg)
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 := · · ·
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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 := · · ·
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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 := · · ·
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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 := · · ·
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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4. New results
Theorem 1. With respect to the ring presentation
H∗(G2) = ∆Z(%3)⊗ ΛZ(%11)⊕ τ2(G2),
where
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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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4. New results
Theorem 2. With respect to the ring presentation
H∗(F4) = ∆Z(%3)⊗ ΛZ(%11, %15, %23)⊕ τ2(F4)⊕ τ3(F4)
where
τ2(F4) = F2[x6]+/
⟨x26
⟩⊗∆F2(%3)⊗ ΛF2(%15, %23)
τ3(F4) = F3[x8]+/
⟨x38
⟩⊗ ΛF3(%3, %11, %15)
where
%23 = x6, x6%11 = 0, x8%23 = 0,
the reduced co–product ψ is given by
{%3, x6, x8} ⊂ P(F4)
ψ(%11) = δ2(ζ5 ⊗ ζ5) + x8 ⊗ %3
ψ(%15) = −δ3(ζ7 ⊗ ζ7),
ψ(%23) = δ3(ζ7 ⊗ ζ7x8 − ζ7x8 ⊗ ζ7).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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4. New results
Theorem 3. With respect to the ring presentation
H∗(E6) = ∆Z(%3)⊗ΛZ(%9, %11, %15, %17, %23)⊕ τ2(E6)⊕ τ3(E6)
where
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,
the reduced co–product ψ is given by
{%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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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)
where
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
,
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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4. New results
the reduced co–product ψ is given by
{%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).
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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)
where
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),
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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4. New results
and where
%23 = x6, %
215 = x30, %
223 = x6
6x10,
x2s%3s−1 = 0, for s = 4, 5
x8%59 = x220C{4,10}, x20%23 = x2
8C{4,10},
x12%59 = 0,
the reduced co–product ψ is given by
{%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);
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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);
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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
26
+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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
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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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 92: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/92.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 93: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/93.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 94: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/94.jpg)
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.
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus
![Page 95: Schubert calculus and cohomology of Lie groupsmasuda/toric/Duan.pdf · Schubert calculus and cohomology of Lie groups Haibao Duan, Institute of Mathematics, CAS Toric Topology 2011](https://reader033.vdocuments.mx/reader033/viewer/2022052023/60381f120bb00860c36d2e61/html5/thumbnails/95.jpg)
Thanks!
Haibao Duan, Institute of Mathematics, CAS Schubert Calculus