Introduction Schubert Calculus on a Grassmann Algebra Equivariant Cohomology of Grassmannians Questions Bibliography Grazie Ph.D. Thesis Schubert Calculus on a Grassmann Algebra Ta´ ıse Santiago Costa Oliveira Research Advisor: Letterio Gatto Politecnico di Torino 27 Marzo 2006 Ta´ ıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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)).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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)).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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)).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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)).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
What is Schubert Calculus on a Grassmann Variety?
As an example,
the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is
Z[`]
(`3),
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!
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
What is Schubert Calculus on a Grassmann Variety?
As an example, the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is
Z[`]
(`3),
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!
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
What is Schubert Calculus on a Grassmann Variety?
As an example, the intersection ring of the grasmannianG (1, C3) ∼= P2 := P(C3) is
Z[`]
(`3),
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!
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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);
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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)) =∞∑
n=0
d1,n+1
n!zn
General (new) combinatorial formulas expressing degrees of topintersections in grassmannian of lines (easily generalizable to other cases);
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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)) =∞∑
n=0
d1,n+1
n!zn
General (new) combinatorial formulas expressing degrees of topintersections in grassmannian of lines (easily generalizable to other cases);
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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;
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
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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:
1 2 3 4 5 6 7−1−2−3−4
A
B
E
C
H
Punti di bloccoPartenza
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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:
1 2 3 4 5 6 7−1−2−3−4
A
B
E
C
H
Punti di bloccoPartenza
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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:
1 2 3 4 5 6 7−1−2−3−4
A
B
E
C
H
Punti di bloccoPartenza
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
What is Schubert Calculus on a Grassmann Variety?Main Goal of the ThesisThe results
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).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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
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 II
The explicit way to phrase that Dt is an A-algebra homomorphism is
Dt(α ∧ β) = Dtα ∧ Dtβ, ∀α, β ∈∧
M (1)
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∑
i=0
Diα ∧ Dh−iβ, ∀α, β ∈∧
M
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∑
i=0
Diα ∧ Dh−iβ, ∀α, β ∈∧
M
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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SCGA - The Theory IV
The SCGA (∧
M,Dt) is based on Leibniz rule
Dh(α ∧ β) =h∑
i=0
Diα ∧ Dh−iβ, ∀α, β ∈∧
M
and on integration by parts
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 ]).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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 ]).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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 ]).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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 ]).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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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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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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Grazie
The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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Grazie
The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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Grazie
The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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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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
and
D1µn =
n+1∑j=1
an+1j µn+1−j .
where:
akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.
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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
and
D1µn =
n+1∑j=1
an+1j µn+1−j .
where:
akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.
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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
and
D1µn =
n+1∑j=1
an+1j µn+1−j .
where:
akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.
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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
and
D1µn =
n+1∑j=1
an+1j µn+1−j .
where:
akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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
and
D1µn =
n+1∑j=1
an+1j µn+1−j .
where:
akh ∈ Ah, ∀k > 0 and all 0 ≤ h ≤ k.
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Simple and Regular SCGAs (over free modules) III
Let D(0)t = 1
1−D1t=∑
i≥0 D i1t
i .
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),
while
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Simple and Regular SCGAs (over free modules) III
Let D(0)t = 1
1−D1t=∑
i≥0 D i1t
i . 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),
while
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Simple and Regular SCGAs (over free modules) III
Let D(0)t = 1
1−D1t=∑
i≥0 D i1t
i . 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),
while
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Simple and Regular SCGAs (over free modules) III
Let D(0)t = 1
1−D1t=∑
i≥0 D i1t
i . 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),
while
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Simple and Regular SCGAs (over free modules) III
Let D(0)t = 1
1−D1t=∑
i≥0 D i1t
i . 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),
while
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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:
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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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Intersection ring of a k − SCGP
The intersection ring of the k-SCGP A∗(∧1+k(M,D1)) is, by definition:
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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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Intersection ring of a k − SCGP
The intersection ring of the k-SCGP A∗(∧1+k(M,D1)) is, by definition:
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)
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Intersection ring of a k − SCGP
The intersection ring of the k-SCGP A∗(∧1+k(M,D1)) is, by definition:
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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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
Intersection ring of a k − SCGP
What can we do out of this algebraic stuff?
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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)))
(2)
is commutative, where ∩ is the capping bilinear map, the upper horizontal mapsends
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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
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?
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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?
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
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
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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Bibliography IV
H. Schubert, Anzahl-Bestimmungen fur lineare Raume beliebigerDimension, Acta. Math., 8 (1886), pp. 97-118.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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Grazie
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
Grazie
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
Grazie
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
QuestionsBibliography
Grazie
Fim
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra