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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);
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
.
.
.
.
.
.
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);
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
.
.
.
.
.
.
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);
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
.
.
.
.
.
.
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);
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
.
.
.
.
.
.
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);
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
.
.
.
.
.
.
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
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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.
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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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
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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
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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
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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
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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
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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
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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 .
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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.
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 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.
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
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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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
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
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
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
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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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.
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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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
QuestionsBibliography
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
QuestionsBibliography
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
QuestionsBibliography
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
QuestionsBibliography
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 (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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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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
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 (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
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 (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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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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
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 (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
IntroductionSchubert Calculus on a Grassmann AlgebraEquivariant Cohomology of Grassmannians
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The TheorySimple and Regular SCGAs (over free modules) IIntersection Theory of Grassmann Bundles
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).
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 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:
A[T]
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)
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:
A[T]
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)
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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:
A[T]
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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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:
A[T]
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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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:
A[T]
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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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
(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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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
(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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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
(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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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
(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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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
(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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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
(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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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-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-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-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-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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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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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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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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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:
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:
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:
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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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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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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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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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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The surprise is that
the same type of “dictionary” found for Grassmann bundles works in thissituation, too.
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The surprise is that
the same type of “dictionary” found for Grassmann bundles works in thissituation, too.
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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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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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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
.
Main theorem for Grassmann bundle
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)))
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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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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:
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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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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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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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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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
n
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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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
n
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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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
n
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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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 ) =
=kX
u=0
X
(mi )
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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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 ) =
=kX
u=0
X
(mi )
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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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 ) =
=kX
u=0
X
(mi )
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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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 ) =
=kX
u=0
X
(mi )
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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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 ) =
=kX
u=0
X
(mi )
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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 ) =
=kX
u=0
X
(mi )
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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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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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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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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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∑
r=0
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 =
=k∑
j=0
µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +
(k∑
r=0
Yir −k∑
r=1
Yr
)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∑
r=0
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 =
=k∑
j=0
µi0 ∧ . . . ∧ µij+1 ∧ . . . µik +
(k∑
r=0
Yir −k∑
r=1
Yr
)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
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
First, one constructs an equilateral triangle of side 7
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
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
Secondly, label the left side with 1000111
1
0
0
0
1
1
1
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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One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
and the right side with 0011011
1
0
0
0
1
1
1 0
0
1
1
0
1
1
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
Finally, one tries to fill the equilateral triangle with the puzzle piecesbelow:
0 0
0
0
00 1 1
1
1
1 1 0 0
1
1
1 0
0 1
0
0
1 1
0 1
1 0
Ordinary pieces
Equivariant piece
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
For example:
1
0
0
0
1
1
1 0
0
1
1
0
1
1
0 1
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
continuing as in the last picture, the possible puzzles having the basislabelled by 1001011
1
0
0
0
1
1
1 0
0
1
1
0
1
1
1 10 10 1 0
0 10 1
are:
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
1 1
1
0
0
0
0
0
1 0
01
01
01
0
0
1 0
0
0
1
1
0
00
00
1
11
1
11
1
10
01
1
1
1
1
1
1 1
0
1
1
1
1 10
11
1
0
1
01
1
10
0
1
0
0
1
11
Weight of the equivariant piece
y1 − y0
y0 y1 y2 y3 y4 y5 y6
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
1
0
0
0
1
1
1 0
0
0
0
0
1
1
0
1
1
1
1
01
0
1
1
1
1
10
0
0
00
0 1
10
0
1
1
1
1
1 1
1
11
01
1
1
1
1
1
1
0
10
0
1
1
00
0
0
1
1 0
1
11
0
0
y0 y1 y2 y3 y4 y5 y6
Weight of the equivariant piece
y2 − y1
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
11
1
0 0
0
0
0
1 0
01
1
01
0
0
0
1 0
0
1
1
1
0
0
0
0
0
1
1 1
1
11
1
10
0 11
1
1
1
1
1
1
0
1
1
1
0
1 0
0
11
0
1
0 11
10
0
1 0
11
1
10
1
Weight of the equivariant piece
y3 − y2
y0 y1 y2 y3 y4 y5 y6
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
One example using Puzzles
Example
Let us compute Sa123 · Sa014 = S1000111 · S0011011, using puzzle rules.
11
1
0 0
0
0
0
1 0
01
1
01
0
0
0
1 0
0
0
1
1
0
0
0
0
0
1
1 1
1
11
1
10
0 11
1
1
1
1
1
1
0
1
1
1
0
1 1
0
1
1
0
1
0 11
10
0
1 0
11
1
10
y4 − y3
Weight of the equivariant piece
y0 y1 y2 y3 y4 y5 y6
Thus, the total contribution of the equivariant pieces in the cycleS1001011 = S124 is:
(y1 − y0) + (y2 − y1) + (y3 − y2) + (y4 − y3) = y4 − y0.
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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.
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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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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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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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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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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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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
Thus,
Gµ123(D) · Gµ
014(D) = Gµ125(D) + Y4 · Gµ
124(D) + (Y1 + Y2 + Y3 + Y4)Y4 · Gµ123(D).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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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
Thus,
Gµ123(D) · Gµ
014(D) = Gµ125(D) + Y4 · Gµ
124(D) + (Y1 + Y2 + Y3 + Y4)Y4 · Gµ123(D).
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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T-Equivariant Cohomology of Grassmannians IEquivariant Pieri’s FormulaOne example via puzzlesOne example using SCGA Theory
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
Thus,
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
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?
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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
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
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
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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
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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
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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
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Bibliography IV
H. Schubert, Anzahl-Bestimmungen fur lineare Raume beliebigerDimension, Acta. Math., 8 (1886), pp. 97-118.
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Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra
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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
Fim
Taıse Santiago Costa Oliveira Ph.D. Thesis Schubert Calculus on a Grassmann Algebra