Equivariant Quantum Cohomology of Homogeneous Spaces

DEDICATION ............................................................................................................................ ii
ACKNOW LEDGEMENTS .................................................................................................... iii
LIST OF FIGURES ................................................................................................................ vi
LIST OF APPENDICES ....................................................................................................... vii
CHAPTER
1. Intro d u ctio n ................................................................................................................... 1
1.1 Motivation ............................................................................................................ 1
1.2 (Imprecise) Statement of results ........................................................................ 3
1.3 Statement of results - for experts ..................................................................... 5
1.3.1 Definitions and notations for general X = G / P ................................. 5
1.3.2 An equivariant quantum Chevalley rule and an algorithm ............... 7
1.3.3 Positivity ............................................................................................. 9
1.3.4 An equivariant quantum Giambelli formula for the Grassmannian . 9
1.4 Structure of the thesis .......................................................................................... 13
2. Prelim inaries ................................................................................................................... 14
2.1 Classical cohomology - establishing notations ................................................... 14
2.1.1 Roots and lengths .................................................................................. 14
2.1.2 Cohomology .......................................................................................... 15
2.1.3 (Schubert) Curves, divisors and degrees ... ........................................... 16
2.2 Equivariant cohomology ....................................................................................... 17
2.2.1 General fa c ts ....................................................................................... 17
2.2.2 Equivariant Schubert calculus on G / P ............................................. 19
2.3 Quantum cohomology .......................................................................................... 23
2.4 Equivariant quantum cohomology of the homogeneous spaces ......................... 25
3. Equivariant quantum Schubert calculus ............................................................... 28
3.1 The equivariant quantum Chevalley rule ........................................................ 28
3.2 Two formulae ...................................................................................................... 33
3.3 An algorithm to compute the EQLR coefficients ............................................ 39
3.3.1 Remarks about the algorithm ............................................................ 42
3.4 Consequences in equivariant cohomology of G/P .............................................. 43
3.5 A brief survey of the algorithms computing the equivariant or quantum
Littlewood-Richardson coefficients ..................................................................... 45
3.6 Appendix - Proof of the Lemma 3.11 ............................................................... 48
4. Positivity in the equivariant quantum Schubert calcu lu s ................................ 52
4.1 Preliminaries ......................................................................................................... 52
4.2 Proof of the positivity Theorem ......................................................................... 55
5. Equivariant quantum cohomology of the Grassmannian ................................... 59
5.1 General facts ......................................................................................................... 59
5.1.1 Definitions and notations for partitions ............................................. 60
5.1.2 Schubert varieties ................................................................................. 61
5.1.3 Equivariant cohomology ..................................................................... 62
5.1.4 Equivariant quantum cohomology ...................................................... 63
5.2 A vanishing property of the EQLR coefficients ................................................ 64
5.3 Equivariant quantum Chevalley-Pieri rule ......................................................... 70
5.4 Relation between the two torus actions ............................................................ 73
5.5 Computation of EQLR coefficients for some “small” Grassmannians ............ 75
5.5.1 The algorithm for the Grassmannian - revisited .............................. 75
5.5.2 Computation of the coefficients for Gr(2,5) .............................. 76
5.5.3 The coefficients for small Grassmannians ................................. 77
5.5.4 Multiplication table for QH^.(Gr(2,4 )) ............................................. 77
6. Polynomial representatives for the equivariant quantum Schubert classes
of the Grassm annian ........................................................................................................ 79
6.1 Factorial Schur functions .................................................................................... 79
6.2 Proof of the formulae .......................................................................................... 84
6.2.1 A characterization of the equivariant quantum cohomology ........... 85
6.2.2 An equivariant quantum Giambelli and presentation ..................... 86
A PPEN DICES ................................................................................................................................. 94
BIBLIOGRAPHY ........................................................................................................................... 116