Ph.D. Thesis Schubert Calculus on a Grassmann...

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Ph.D. ThesisSchubert Calculus on a Grassmann Algebra

Taıse Santiago Costa Oliveira

Research Advisor: Letterio Gatto

Politecnico di Torino

27 Marzo 2006

Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra

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What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results

What is a Schubert Calculus on a Grassmann Variety?

Given m · p subspaces of dimension p in general position in a complexvector space Cm+p, how many subspaces of dimension m intersect allthese m · p subspaces nontrivially?

This question is a classical example of a Schubert calculus question(Schubert (1848–1911), Pieri (1860–1913), Giambelli (1879–1953)).

To be able to answer such questions, one needs to gain a concreteunderstanding of the structure of the cohomology ring of thecorresponding variety (in the example above, the GrassmannianG (m, Cm+p)).

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What is Schubert Calculus on a Grassmann Variety?

As an example,

the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is

where ` is the “class of a hyperplane” (in this case a hyperplane in P2 isa line). The relation `3 = 0 means that:

Three special lines can intersect But the general ones do not!

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As an example, the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is

where ` is the “class of a hyperplane” (in this case a hyperplane in P2 isa line).

The relation `3 = 0 means that:

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As an example, the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is

where ` is the “class of a hyperplane” (in this case a hyperplane in P2 isa line). The relation `3 = 0 means that:

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Main Goal of the Thesis

Proposing a (new) axiomatic approach able to describe, within a unifiedframework, different kind of intersection theories living on grassmannians,such as the classical, the (small) quantum and the equivariant one.

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Main Goal of the Thesis

Proposing a (new) axiomatic approach able to describe, within a unifiedframework, different kind of intersection theories living on grassmannians,such as the classical, the (small) quantum and the equivariant one.

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The results I

Using such an axiomatic theory we can get

The extension of Gatto’s interpretation of Schubert Calculus forgrassmannians ( see [3] [Schubert Calculus via Hasse–Schmidt

Derivations, Asian J. Math., 9, No. 3, (2005), 315–322]) to theintersection theory of Grassmann bundles;

This allows:

New proofs of classical formulas

expressing the degree of Schubert varieties;

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The results I

This allows:

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The results I

This allows:

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The results I

This allows:

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The results I

This allows:

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The results II

New formulas expressing solutions of enumerative problems, such as:

Number Nd of rational curves in P3 of degree d having flexes at 2d − 6prescribed points (suggested by K. Ranestad);

Nd =Xτ∈S4(ni )

(−1)|τ |� 2(d − 3)

n0, n1, . . . , n9

where the ni satisfy:

8>><>>:

2n0 + n1 + n3 + n6 = d − 3 + τ(0)2n2 + n1 + n4 + n7 = d − 4 + τ(1)2n5 + n3 + n4 + n8 = d − 5 + τ(2)2n9 + n4 + n6 + n7 = d − 6 + τ(3)

Putting this formula in R (orMathematica c©), we obtain the follow table

d Nd0 11 02 13 54 1265 33966 1146757 44307128 1907205309 894218863210 449551230102

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The results II

Nd =Xτ∈S4(ni )

(−1)|τ |� 2(d − 3)

n0, n1, . . . , n9

8>><>>:

d Nd0 11 02 13 54 1265 33966 1146757 44307128 1907205309 894218863210 449551230102

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The results II

Nd =Xτ∈S4(ni )

(−1)|τ |� 2(d − 3)

n0, n1, . . . , n9

8>><>>:

d Nd0 11 02 13 54 1265 33966 1146757 44307128 1907205309 894218863210 449551230102

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The results II

Nd =Xτ∈S4(ni )

(−1)|τ |� 2(d − 3)

n0, n1, . . . , n9

8>><>>:

d Nd0 11 02 13 54 1265 33966 1146757 44307128 1907205309 894218863210 449551230102

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The results II

Nd =Xτ∈S4(ni )

(−1)|τ |� 2(d − 3)

n0, n1, . . . , n9

8>><>>:

d Nd0 11 02 13 54 1265 33966 1146757 44307128 1907205309 894218863210 449551230102

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The results III

A generating function for the degrees d1,n+1 of the grassmannians of linesin terms of modified Bessel’s functions

F (z) = e2z(I0(2z)− I1(2z)) =∞∑

d1,n+1

General (new) combinatorial formulas expressing degrees of topintersections in grassmannian of lines (easily generalizable to other cases);

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The results III

A generating function for the degrees d1,n+1 of the grassmannians of linesin terms of modified Bessel’s functions

F (z) = e2z(I0(2z)− I1(2z)) =∞∑

d1,n+1

General (new) combinatorial formulas expressing degrees of topintersections in grassmannian of lines (easily generalizable to other cases);

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The results IV

New interpretation of a traffic game [14]( Niederhausen, [Catalan

traffic at the beach - E.J.C., 9, ] R33 (2002), 1–17]) in terms of degreesof top intersections in grassmannian of lines;

We prove the following:

Theorem

For all 0 ≤ m ≤ n, the number of paths joining (0, 0) to (m, n) ∈ ς isequal to the number of lines in Pn+1 incident 2m linear subspaces ofcodimension 2 and n −m subspaces of codimension 3 in general positionin Pn+1.

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The results IV

Theorem

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The results IV

Let ς a city map with traffic constraints depicted below. How manydistincts paths joining (0, 0) to (m, n) are there in ς?

We prove thefollowing:

Theorem

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The results IV

1 2 3 4 5 6 7−1−2−3−4

Punti di bloccoPartenza

Theorem

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The results IV

1 2 3 4 5 6 7−1−2−3−4

Theorem

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The results IV

1 2 3 4 5 6 7−1−2−3−4

Theorem

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The results IV

Theorem

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The results IV

Theorem

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The results V

However, we guess that the most important achievement consists in:

A flexible description of the equivariant cohomology ofgrassmannians, alternative to that using puzzle techniques [9] (Knutson-Tao [Puzzles and (equivariant) cohomology of Grassmannians,

Duke Math. J. 119, no. 2 (2003), 221260]), allowing us to answersome questions asked in the literature such as

Finding an equivariant Giambelli’s formula

and even more important

the full set of equivariant Pieri formulas(while only that for codimension 1 was previously explicitly known).

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The results V

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The results V

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The results V

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The results V

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The results V

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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles

SCGA - The Theory I

Definition

A SCGA is a pair (∧

M,Dt), where M is a module over an integralZ-algebra A and

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

is an A-algebra homomorphism whose coefficients Di ∈ EndA(∧

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

D0 is an automorphism.

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SCGA - The Theory I

Definition

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

M,Dt),

where M is a module over an integralZ-algebra A and

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

M,Dt), where M is a module over an integralZ-algebra A

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory I

Definition

Dt :=∑i≥0

Di ti :∧

M −→∧

M[[t]],

M) aresuch that

Di ◦ Dj = Dj ◦ Di , ∀i , j ∈ N;

Di (∧r M) ⊆

∧r M;

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SCGA - The Theory II

The explicit way to phrase that Dt is an A-algebra homomorphism is

Dt(α ∧ β) = Dtα ∧ Dtβ, ∀α, β ∈∧

which is said to be

the fundamental equation of Schubert Calculus Dt .

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SCGA - The Theory III

The fundamental equation is equivalent to:

Dh(α ∧ β) =h∑

Diα ∧ Dh−iβ, ∀α, β ∈∧

which is the hth order Leibniz rule, holding for all h ≥ 0.

The set of all Dt defining a Schubert Calculus on∧

M form a groupSt(∧

M) with respect to the product

Dt ∗ Et =∑h≥0

∑i+j=h

(Di ◦ Ej)th.

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SCGA - The Theory III

The fundamental equation is equivalent to:

Dh(α ∧ β) =h∑

which is the hth order Leibniz rule, holding for all h ≥ 0.

The set of all Dt defining a Schubert Calculus on∧

M form a groupSt(∧

M) with respect to the product

Dt ∗ Et =∑h≥0

∑i+j=h

(Di ◦ Ej)th.

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SCGA - The Theory IV

The SCGA (∧

M,Dt) is based on Leibniz rule

and on integration byparts

Dhα ∧ β =∑i≥0

(−1)iDh−i (α ∧ D iβ)

where Dt is the formal inverse of Dt : Dt =∑

i≥0(−1)iD i ti .

Leibniz rule and integration by parts are the very abstract counterparts ofPieri and Giambelli formula respectively (see also Laksov-Thorup [12][A

Determinantal Formula for the Exterior Powers of the Polynomial Ring -

Preprint, 2005 ]).

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The SCGA (∧

M,Dt) is based on Leibniz rule

Dh(α ∧ β) =h∑

and on integration by parts

Dhα ∧ β =∑i≥0

i≥0(−1)iD i ti .

Preprint, 2005 ]).

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The SCGA (∧

M,Dt) is based on Leibniz rule and on integration byparts

Dhα ∧ β =∑i≥0

i≥0(−1)iD i ti .

Preprint, 2005 ]).

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The SCGA (∧

Dhα ∧ β =∑i≥0

i≥0(−1)iD i ti .

Preprint, 2005 ]).

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The SCGA (∧

Dhα ∧ β =∑i≥0

i≥0(−1)iD i ti .

Preprint, 2005 ]).

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SCGA - The Theory V

In this thesis we have deduced properties common to any SCGA(∧

M,Dt). All of them are consequence of the fundamental equation

Dt(α ∧ β) = Dtα ∧ Dtβ.

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Simple and Regular SCGAs

In this exposition we shall restrict to special kinds of SCGAs, translatinga big part of work already done by Laksov and Thorup [12] [loc. cit.].More precisely:

A will be a graded ring of characteristic 0

A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

(for instance you may think to A = Z[X1, . . . ,Xm]).

M a free A-module generated by (µ0, µ1, . . . , µn), for some n ∈ N.

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In this exposition we shall restrict to special kinds of SCGAs, translatinga big part of work already done by Laksov and Thorup [12] [loc. cit.].

More precisely:

A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

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A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

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A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

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A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

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A := A0 ⊕ A1 ⊕ . . . , A0∼= Z

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Simple and Regular SCGAs (over free modules) II

Let D1 : M −→ M be the unique endomorphism of M such that:

D1µi = µi+1 +

i+1∑j=1

ai+1j µi+1−j , 0 ≤ i ≤ n − 1

D1µn =

n+1∑j=1

an+1j µn+1−j .

where:

akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.

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D1µi = µi+1 +

i+1∑j=1

ai+1j µi+1−j , 0 ≤ i ≤ n − 1

D1µn =

n+1∑j=1

an+1j µn+1−j .

where:

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D1µi = µi+1 +

i+1∑j=1

ai+1j µi+1−j , 0 ≤ i ≤ n − 1

D1µn =

n+1∑j=1

an+1j µn+1−j .

where:

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D1µi = µi+1 +

i+1∑j=1

ai+1j µi+1−j , 0 ≤ i ≤ n − 1

D1µn =

n+1∑j=1

an+1j µn+1−j .

where:

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D1µi = µi+1 +

i+1∑j=1

ai+1j µi+1−j , 0 ≤ i ≤ n − 1

D1µn =

n+1∑j=1

an+1j µn+1−j .

where:

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Simple and Regular SCGAs (over free modules) III

Let D(0)t = 1

1−D1t=∑

i≥0 D i1t

It can be easily proven that

There exists a unique SCGA (∧

M,Dt) such that Dtm = D(0)t m, ∀m ∈ M.

Such SCGA will be denoted by∧(M,D1),

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Let D(0)t = 1

1−D1t=∑

i≥0 D i1t

i . It can be easily proven that

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Let D(0)t = 1

1−D1t=∑

i≥0 D i1t

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Let D(0)t = 1

1−D1t=∑

i≥0 D i1t

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Let D(0)t = 1

1−D1t=∑

i≥0 D i1t

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Simple and Regular SCGAs (over free modules IV)

1+k∧(M,D1) := (

1+k∧M,Dt |∧1+k M

is said to be a

Schubert Calculus on the (1 + k)th-Grassmann power (k-SCGP).

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Simple and Regular SCGAs (over free modules IV)

Let Z[T] := Z[T1,T2, . . .] and let A[T] := Z[T]⊗Z A. For each k ≥ 0,consider the map

evD,µ0∧µ1∧...∧µk : A[T] −→∧1+k M

P 7−→ P(D) · µ0 ∧ µ1 ∧ . . . ∧ µk .

Here, P(D) is the endomorphism of∧1+k M gotten by “substituting”

Ti = Di into the polynomial P.

Theorem

The map evD,µ0∧µ1∧...∧µk is surjective.

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Intersection ring of a k − SCGP

The intersection ring of the k-SCGP A∗(∧1+k(M,D1)) is, by definition:

ker(evD,µ0∧µ1∧...∧µk ).

Let G ∈ A[T] such that

G (D)µ0 ∧ µ1 ∧ . . . ∧ µk = µi0 ∧ µi1 ∧ . . . ∧ µik

G will be said a Giambelli’s polynomial of µi0 ∧ µi1 ∧ . . . ∧ µik . It’s uniquemodulo ker(evD,µ0∧µ1∧...∧µk ). Abusing terminology G(D) will be saidthe Giambelli’s polynomial of µi0 ∧ µi1 ∧ . . . ∧ µik .

Notation : Gµi0,i1,...,ik

(D) := G (D)

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G (D)µ0 ∧ µ1 ∧ . . . ∧ µk = µi0 ∧ µi1 ∧ . . . ∧ µik

(D) := G (D)

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G (D)µ0 ∧ µ1 ∧ . . . ∧ µk = µi0 ∧ µi1 ∧ . . . ∧ µik

(D) := G (D)

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G (D)µ0 ∧ µ1 ∧ . . . ∧ µk = µi0 ∧ µi1 ∧ . . . ∧ µik

G will be said a Giambelli’s polynomial of µi0 ∧ µi1 ∧ . . . ∧ µik . It’s uniquemodulo ker(evD,µ0∧µ1∧...∧µk ).

Abusing terminology G(D) will be saidthe Giambelli’s polynomial of µi0 ∧ µi1 ∧ . . . ∧ µik .

(D) := G (D)

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G (D)µ0 ∧ µ1 ∧ . . . ∧ µk = µi0 ∧ µi1 ∧ . . . ∧ µik

(D) := G (D)

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What can we do out of this algebraic stuff?

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Intersection Theory of Grassmann Bundles I

First of all classical intersection theory on Grassmann bundles isrecovered by this model!

One of our main theorems is the generalization of Gatto’s result, i.e. thatthe intersection theory of a Grassmann bundle pk : Gk(P(E )) −→ Xassociated to the vector bundle p : E −→ X over a smooth connectedvariety X can be described via the SCGA language:

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Intersection Theory of Grassmann Bundles I

First of all classical intersection theory on Grassmann bundles isrecovered by this model!

One of our main theorems is the generalization of Gatto’s result, i.e. thatthe intersection theory of a Grassmann bundle pk : Gk(P(E )) −→ Xassociated to the vector bundle p : E −→ X over a smooth connectedvariety X can be described via the SCGA language:

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Intersection Theory of Grassmann Bundles II

Theorem: The diagram

A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

A∗(Gk(P(E)))⊗A∗(X ) A∗(Gk(P(E)))∩−→ A∗(Gk(P(E)))

is commutative, where ∩ is the capping bilinear map, the upper horizontal mapsends

(Dh, εi0 ∧ εi1 ∧ . . . ∧ εik ) 7→ Dh(εi0 ∧ εi1 ∧ . . . ∧ εik ),

whereC : Di 7−→ ci (Qk − p∗k E),

and whereεi0 ∧ εi1 ∧ . . . ∧ εik 7−→∆(εi0 ∧ εi1 ∧ . . . ∧ εik ),

∆(εi0 ∧ εi1 ∧ . . . ∧ εik ) = ∆(i0,i1,...,ik )(ct(Qk − p∗k E)) ∩ [Gk(P(E))].

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A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

is commutative,

where ∩ is the capping bilinear map, the upper horizontal mapsends

and whereεi0 ∧ εi1 ∧ . . . ∧ εik 7−→ ∆(εi0 ∧ εi1 ∧ . . . ∧ εik ),

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A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

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A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

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A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

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A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory

T-Equivariant Cohomology of Grassmannians I

Let X be a complex smooth projective variety acted on by

T := (C∗)1+p.

and let

ET −→ BT

the universal T -principal bundle. Where:

ET = C∞ \ {0} × . . .× C∞ \ {0}︸︷︷︸1+p times

and BT = P∞ × . . .× P∞︸︷︷︸1+p times

If T acts on X , then T ′ := (S1)1+p acts on X as well.

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T := (C∗)1+p.

and let

ET −→ BT

ET = C∞ \ {0} × . . .× C∞ \ {0}︸︷︷︸1+p times

and BT = P∞ × . . .× P∞︸︷︷︸1+p times

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T := (C∗)1+p.

and let

ET −→ BT

ET = C∞ \ {0} × . . .× C∞ \ {0}︸︷︷︸1+p times

and BT = P∞ × . . .× P∞︸︷︷︸1+p times

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T := (C∗)1+p.

and let

ET −→ BT

ET = C∞ \ {0} × . . .× C∞ \ {0}︸︷︷︸1+p times

and BT = P∞ × . . .× P∞︸︷︷︸1+p times

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T := (C∗)1+p.

and let

ET −→ BT

ET = C∞ \ {0} × . . .× C∞ \ {0}︸︷︷︸1+p times

and BT = P∞ × . . .× P∞︸︷︷︸1+p times

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T-Equivariant Cohomology of Grassmannians II

Define the T -equivariant cohomology of X as being

H∗T (X ) := H∗(X ×T ET ) and H∗

T ′(X ) := H∗(X ×T ′ ET ′).

Since S1 is a deformation retract of C∗, basic results ensure that

H∗T (X ) = H∗

T ′(X ).

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H∗T (X ) := H∗(X ×T ET ) and H∗

T ′(X ) := H∗(X ×T ′ ET ′).

H∗T (X ) = H∗

T ′(X ).

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H∗T (X ) := H∗(X ×T ET ) and H∗

T ′(X ) := H∗(X ×T ′ ET ′).

H∗T (X ) = H∗

T ′(X ).

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T-Equivariant Chow Group I

Hence, from now on we shall only deal with T -equivariant intersectiontheory.

It will not be defined here, but:

a result of A. Bialynicki-Birula ([1]) ensures us that there is a naturalisomorphism

A∗T (Gk(Pn)) ∼= H∗T (Gk(Pn)),

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Hence, from now on we shall only deal with T -equivariant intersectiontheory. It will not be defined here, but:

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T-Equivariant Chow Group II

Recall that A∗T (Pn) is a is a free module of rank n + 1 over

A = A∗T (pt) = Z[y0, y1, . . . , yp].

Suppose there is a regular 0-SCGP (M,D1), where

M = Aµ0 ⊕ Aµ1 ⊕ . . .⊕ Aµn,

such that:

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A = A∗T (pt) = Z[y0, y1, . . . , yp].

M = Aµ0 ⊕ Aµ1 ⊕ . . .⊕ Aµn,

such that:

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A = A∗T (pt) = Z[y0, y1, . . . , yp].

M = Aµ0 ⊕ Aµ1 ⊕ . . .⊕ Aµn,

such that:

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T-Equivariant Chow Group III

M is a free A∗T (Pn)-module of rank 1 generated by µ0;

The rings A∗T (Pn) and A∗(M,D1) ∼= A[T]/(ker(evD,µ0) areisomorphic, and the following diagram

A∗T (Pn)⊗A M −→ My yA∗(M,D1)⊗A M −→ M

is commutative, the vertical arrows being isomorphisms (the secondone is the identity).

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T-Equivariant Chow Group IV

The surprise is that

the same type of “dictionary” found for Grassmann bundles works in thissituation, too.

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T-Equivariant Chow Group IV

The surprise is that

the same type of “dictionary” found for Grassmann bundles works in thissituation, too.

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T-Equivariant Chow Group V

T -Equivariant Schubert calculus on Gk(Pn) is described by∧1+k(M,D1)

in the following sense:

Theorem

The following “equivariant dictionary” holds

A∗T (Gk(Pn))⊗A

∧1+k M −→∧1+k M

ιk ⊗ 1y y1

A∗(∧1+k(M,D1))⊗A

∧1+k M −→∧1+k M

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Theorem

A∗T (Gk(Pn))⊗A

∧1+k M −→∧1+k M

ιk ⊗ 1y y1

A∗(∧1+k(M,D1))⊗A

∧1+k M −→∧1+k M

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Theorem

A∗T (Gk(Pn))⊗A

∧1+k M −→∧1+k M

ιk ⊗ 1y y1

A∗(∧1+k(M,D1))⊗A

∧1+k M −→∧1+k M

Main theorem for Grassmann bundle

A∗(V1+k(M, D1))⊗A∗(X )

V1+k M −→V1+k M??yC⊗∆??y∆

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Theorem

A∗T (Gk(Pn))⊗A

∧1+k M −→∧1+k M

ιk ⊗ 1y y1

A∗(∧1+k(M,D1))⊗A

∧1+k M −→∧1+k M

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T-Equivariant Chow Group VI

Idea of the proof

One uses 3 facts:

For each m ≥ 0 and for each 0 ≤ i ≤ 2m,H i

T (Gk(Pn)) = H i (EmT ×T Gk(Pn))(i.e. the cohomology of associated bundles to EmT −→ BmTapproximate the equivariant cohomology);

EmT ×T Gk(Pn) −→ BmT ,

is the Grassmann bundle Gk(P(Em)) −→ (Pm)1+p, where Em −→ Xis the holomorphic vector bundle EmT ×T C1+n −→ BmT associatedto EmT −→ BmT .

Since BmT = (Pm)1+p has a cellular decomposition Fulton ( [2], p.378) ensures that : H∗(Gk(P(Em))) = A∗(Gk(P(Em)))

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Idea of the proof

One uses 3 facts:

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Idea of the proof

One uses 3 facts:

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Idea of the proof

One uses 3 facts:

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Idea of the proof

One uses 3 facts:

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T-Equivariant Chow Group VII

Idea of the proof

To end the proof one then applies the main theorem for Grassmannbundles in the diagram:

A∗(Gk(P(Em)))⊗Am

∧1+k M(m)∩−→

∧1+k M(m)

ιk,m ⊗ 1y y1

A∗(∧1+k(M(m),D1,m))⊗Am

∧1+k M(m) −→∧1+k M(m)

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A description of the Equivariant Cohomology ofGrassmannians without puzzles

If p = n then we have:

A description of the Equivariant Cohomology of Grassmannians withoutPuzzles (Knutson and Tao [loc. cit.]).

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A description of the Equivariant Cohomology ofGrassmannians without puzzles

If p = n then we have:

A description of the Equivariant Cohomology of Grassmannians withoutPuzzles (Knutson and Tao [loc. cit.]).

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A description of the Equivariant Cohomology ofGrassmannians without puzzles - The model I

Let D1 : M −→ M is the unique A-linear map such that:

D1µj = µj+1 + (yj − y0)µ

j , 0 ≤ j < n and

D1µn = (yn − y0)µ

and let (M,D1) be the corresponding 0-SCGP.

Proposition

There is a ring isomorphism ι : A∗(M,D1) −→ H∗T (Pn) given by

D1 7→ S1 .

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D1µj = µj+1 + (yj − y0)µ

j , 0 ≤ j < n and

Proposition

D1 7→ S1 .

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D1µj = µj+1 + (yj − y0)µ

j , 0 ≤ j < n and

Proposition

D1 7→ S1 .

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A description of the Equivariant Cohomology ofGrassmannians without puzzles - The model II

Thus, if∧1+k(M,D1) is the k-SCGP associated to this 0-SCGP, the

main theorem for equivariant cohomology says that:

A∗(1+k∧

(M,D1)) ∼= A∗T (Gk(Pn)) = H∗T (Gk(Pn)).

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Equivariant Pieri’s Formula I

Theorem

Let l ≥ 0, µi0 ∧ µi1 ∧ . . . ∧ µik ∈V1+k M and Yi = yi − y0.

Pieri’s formula for T-equivariant cohomology of grassmannians holds:

Dl(µi0 ∧ . . . ∧ µik ) =

hu(Yi0 , . . . , Yi0+m0 , . . . , Yik , . . . , Yik+mk )µi0+m0 ∧ . . . ∧ µik+mk

where (mi ) ∈ P(I , l − u) and

P(I , l − u) is the set of all (1 + k)-tuples (mi ) ∈ N1+k such that:

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Theorem

Dl(µi0 ∧ . . . ∧ µik ) =

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Theorem

Dl(µi0 ∧ . . . ∧ µik ) =

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Theorem

Dl(µi0 ∧ . . . ∧ µik ) =

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Theorem

Dl(µi0 ∧ . . . ∧ µik ) =

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Theorem

Dl(µi0 ∧ . . . ∧ µik ) =

0 ≤ i0 ≤ i0 + m0 < i1 ≤ i1 + m1 < i2 ≤ i2 + m2 < . . . < ik

with m0 + m1 + m2 + . . . + mk = l − u.

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Equivariant Pieri’s Formula II - Example

Example: Using our Pieri’s formula.

Let us compute the coefficient of

µ2 ∧ µ3 ∧ µ7

in the expansion of D3(µ2 ∧ µ3 ∧ µ5).

Since wt(D3(µ2 ∧ µ3 ∧ µ5)) = (2− 0) + (3− 1) + (5− 2) + 3 = 10, such

a coefficient is a polynomial of degree 1 because wt(µ2 ∧ µ3 ∧ µ7) = 9.By our Pieri formula, this is given by:

h1(Y2,Y3,Y5,Y6,Y7) = Y2+Y3+Y5+Y6+Y7 = y2+y3+y5+y6+y7−5y0.

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µ2 ∧ µ3 ∧ µ7

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µ2 ∧ µ3 ∧ µ7

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µ2 ∧ µ3 ∧ µ7

a coefficient is a polynomial of degree 1 because wt(µ2 ∧ µ3 ∧ µ7) = 9.

By our Pieri formula, this is given by:

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µ2 ∧ µ3 ∧ µ7

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Equivariant Pieri’s Formula III

In particular Pieri’s rule for codimension 1 subvarieties, is given by

D1(µi0 ∧ . . . ∧ µik ) =

k∑j=0

µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +k∑

Yik µi0 ∧ . . . ∧ µik

Knutson and Tao, in [9] [loc. cit.], computes a Pieri’s formula forcodimension 1 subvarieties. It can be recovered within our formalism asfollows:

Gµ0,1,...,k−1,k+1(D)µi0 ∧ . . . ∧ µik =

µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +

Yir −k∑

)µi0 ∧ . . . ∧ µik

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Equivariant Pieri’s Formula III

In particular Pieri’s rule for codimension 1 subvarieties, is given by

D1(µi0 ∧ . . . ∧ µik ) =

k∑j=0

µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +k∑

Yik µi0 ∧ . . . ∧ µik

Knutson and Tao, in [9] [loc. cit.], computes a Pieri’s formula forcodimension 1 subvarieties. It can be recovered within our formalism asfollows:

Gµ0,1,...,k−1,k+1(D)µi0 ∧ . . . ∧ µik =

µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +

Yir −k∑

)µi0 ∧ . . . ∧ µik

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Equivariant Giambelli’s formula

The equivariant Giambelli formula instead is given by integration by parts.

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One example using Puzzles

Example

Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.

Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

First, one constructs an equilateral triangle of side 7

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

Grazie

Example

Secondly, label the left side with 1000111

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

Grazie

Example

and the right side with 0011011

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

Finally, one tries to fill the equilateral triangle with the puzzle piecesbelow:

00 1 1

1 1 0 0

Ordinary pieces

Equivariant piece

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

For example:

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

continuing as in the last picture, the possible puzzles having the basislabelled by 1001011

1 10 10 1 0

0 10 1

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

Grazie

Example

Weight of the equivariant piece

y1 − y0

y0 y1 y2 y3 y4 y5 y6

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

Grazie

Example

y0 y1 y2 y3 y4 y5 y6

y2 − y1

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

y3 − y2

y0 y1 y2 y3 y4 y5 y6

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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Example

y4 − y3

y0 y1 y2 y3 y4 y5 y6

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

Grazie

Example

(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.

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One example using SCGA Theory I

Let us compute the product Gµ123(D) · Gµ

014(D) using SCGA Theory. Todo this, we observe that:

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 = Gµ123(D)µ0 ∧ µ1 ∧ µ4

To speed up computations, we shall use canonical bases ε = (ε0, ε1, . . .).One has:

µ1 ∧ µ2 ∧ µ3 = ε1 ∧ ε2 ∧ ε3 and

µ0 ∧ µ1 ∧ µ4 = ε0 ∧ ε1 ∧ ε4 − e1(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε3 +

+e2(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε2.

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One example using SCGA Theory II

Then we must compute:

Gε123(D) · (ε0 ∧ ε1 ∧ ε4 − e1(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε3 + e2(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε2)

But G ε123(D) = D3 where D3 is the endomorphism defined (in the

canonical basis) by D3(εi ∧ εj ∧ εk) = εi+1 ∧ εj+1 ∧ εk+1. Therefore

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

= D3(ε0 ∧ ε1 ∧ ε4 − e1(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε3 + e2(Y1, Y2, Y3)ε0 ∧ ε1 ∧ ε2) =

= ε1 ∧ ε2 ∧ ε5 − e1(Y1, Y2, Y3)ε1 ∧ ε2 ∧ ε4 + e2(Y1, Y2, Y3)ε1 ∧ ε2 ∧ ε3 =

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Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

Grazie

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

Grazie

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

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One example using SCGA Theory III

Returning to the original basis∧3

µ, one easily gets:

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

= µ1 ∧ µ2 ∧ µ5 +�h1(Y1, Y2, Y3, Y4)− e1(Y1, Y2, Y3)

�µ1 ∧ µ2 ∧ µ4 +

+�h2(Y1, Y2, Y3, Y4)− e1(Y1, Y2, Y3)h1(Y1, Y2, Y3) + e2(Y1, Y2, Y3)

�µ1 ∧ µ2 ∧ µ3 =

= µ1 ∧ µ2 ∧ µ5 + Y4µ1 ∧ µ2 ∧ µ4 + (Y1 + Y2 + Y3 + Y4)Y4µ

1 ∧ µ2 ∧ µ3 =

=�Gµ

125(D) + Y4Gµ124(D) + (Y1 + Y2 + Y3 + Y4)Y4G

µ123(D)

�µ0 ∧ µ1 ∧ µ2

Gµ123(D) · Gµ

014(D) = Gµ125(D) + Y4 · Gµ

124(D) + (Y1 + Y2 + Y3 + Y4)Y4 · Gµ123(D).

Grazie

Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

= µ1 ∧ µ2 ∧ µ5 +�h1(Y1, Y2, Y3, Y4)− e1(Y1, Y2, Y3)

�µ1 ∧ µ2 ∧ µ4 +

�µ1 ∧ µ2 ∧ µ3 =

= µ1 ∧ µ2 ∧ µ5 + Y4µ1 ∧ µ2 ∧ µ4 + (Y1 + Y2 + Y3 + Y4)Y4µ

1 ∧ µ2 ∧ µ3 =

=�Gµ

125(D) + Y4Gµ124(D) + (Y1 + Y2 + Y3 + Y4)Y4G

µ123(D)

�µ0 ∧ µ1 ∧ µ2

Gµ123(D) · Gµ

014(D) = Gµ125(D) + Y4 · Gµ

124(D) + (Y1 + Y2 + Y3 + Y4)Y4 · Gµ123(D).

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Gµ123(D) · Gµ

014(D)µ0 ∧ µ1 ∧ µ2 =

= µ1 ∧ µ2 ∧ µ5 +�h1(Y1, Y2, Y3, Y4)− e1(Y1, Y2, Y3)

�µ1 ∧ µ2 ∧ µ4 +

�µ1 ∧ µ2 ∧ µ3 =

= µ1 ∧ µ2 ∧ µ5 + Y4µ1 ∧ µ2 ∧ µ4 + (Y1 + Y2 + Y3 + Y4)Y4µ

1 ∧ µ2 ∧ µ3 =

=�Gµ

125(D) + Y4Gµ124(D) + (Y1 + Y2 + Y3 + Y4)Y4G

µ123(D)

�µ0 ∧ µ1 ∧ µ2

Gµ123(D) · Gµ

014(D) = Gµ125(D) + Y4 · Gµ

124(D) + (Y1 + Y2 + Y3 + Y4)Y4 · Gµ123(D).

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Questions I

Grassmannians are just a very special case of a flag variety. Thecohomology of flag varieties has been recently investigated by Laksov andThorup using the powerful framework of splitting algebras. Such aframework is related with our group St(

∧M). If (1 + D1t), . . . , (1 + Dnt)

are SCGA on∧

M such that (1 + D1t) . . . (1 + Dnt) = 1, thenD1, . . . ,Dn generate the cohomology of the complete flag variety Fl(Cn).One needs to understand the constant structures of A∗(Fl(Cn))(associated to a given basis) generalizing the case of grassmannians.

How to spell the cohomology of the flag varieties in terms of suitableoperators generalizing derivation for grassmannians?

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Questions I

∧M). If (1 + D1t), . . . , (1 + Dnt)

are SCGA on∧

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Questions I

∧M). If (1 + D1t), . . . , (1 + Dnt)

are SCGA on∧

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Questions II

The quantum cohomology of Pn defines a 0-SCGP in our sense. Does thecorresponding k-SCGP

∧k(M,D1) translates the quantum cohomologyof Gk(Pn)?

There is an easy test to do. In fact QH∗(P3) and QH∗(G1(P3)) areexplicitly known, and one is left just with a check (I have not done yet!).If the answer were affirmative one would have a relationship betweenenumerative geometry of rational curves and enumerative geometry ofruled surfaces.

Relationship of our model with the theory of λ-rings.

Relationships with Control theory.

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Questions II

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Questions II

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Bibliography I

A. Bia lynicki-Birula, Some theorems on actions of algebraic groups, Ann.of Math. (2), v. 98, 1973, 480–497.

W. Fulton, Intersection Theory,Springer-Verlag, 1984.

L. Gatto, Schubert Calculus via Hasse–Schmidt Derivations, AsianJ. Math., 9, No. 3, (2005), 315–322.

L. Gatto, Schubert Calculus: an Algebraic Introduction, 25 Coloq. Bras.de Mat., Inst. de Mat. Pura Apl. (IMPA), Rio de Janeiro, 2005.

L. Gatto, T. Santiago, Schubert Calculus on Grassmann Algebra I: GeneralTheory., in preparation, 2006.

L. Gatto, T. Santiago, Schubert Calculus on Grassmann Algebra II:Equivariant Cohomology of Grassmannians, in preparation, 2006

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Bibliography II

M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology,Koszul duality, and the localization theorem, Invent. Math. 131 (1998),25–83.

G. Z. Giambelli, Risoluzione del problema degli spazi secanti, Mem. R.Accad. Torino 52 (1902), 171–211.

A. Knutson, T. Tao, Puzzles and (equivariant) cohomology ofGrassmannians, Duke Math. J. 119, no. 2 (2003), 221260.

S. L. Kleiman, D. Laksov, Schubert Calculus, Amer. Math. Monthly 79,(1972), 1061–1082.

V. Lakshmibai, R. N. Raghavan, P. Sankaran, Equivariant Giambelli anddeterminantal restrictiction formulas for the Grassmannian, Pure Appl.Math. Quarterly (special issue in honour of McPherson on his 60thbirthday, to appear arXiv:mathAG/0506015.

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Bibliography III

D. Laksov, A. Thorup, A Determinantal Formula for the Exterior Powersof the Polynomial Ring, Preprint, 2005 (available from the Authors uponrequest).

D. Laksov, A. Thorup, Universal Splitting Algebras and IntersectionTheory of Flag Schemes, Private Communication, 2004.

H. Niederhausen, Catalan Traffic at the Beach, The Eletr. J. of Combin.,9 (2002), ]R32.

S. Robinson, A Pieri-type formula for H∗T (SLn(C)/B), J Algebra 249,

(2002), 38–58.

T. Santiago C. Oliveira, Degrees of Grassmannians of Lines, Atti Acc. Sci.di Torino, 2005, to appear.

T. Santiago C. Oliveira, “Catalan Traffic” and Integrals on theGrassmannian of Lines, Dip. di Mat. Politecnico di Torino, Rapp. int.n.35, december 2005.

Grazie

Bibliography IV

H. Schubert, Anzahl-Bestimmungen fur lineare Raume beliebigerDimension, Acta. Math., 8 (1886), pp. 97-118.

Grazie

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