Table Of ContentEquivariant quantum cohomology of homogeneous
spaces
by
Constantin Leonardo Mihalcea
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mathematics)
in The University of Michigan
2005
Doctoral Committee:
Professor William E. Fulton, Chair
Professor Sergey Fomin
Professor Robert K. Lazarsfeld
Assistant Professor Jamie Tappenden
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UMI Number: 3186707
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To my parents (Parin^ilor mei)
ii
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ACKNO W LED GEM ENTS
Mathematically, I have benefitted from conversations with many people. Among
them I would like to mention Arend Bayer, Gilberto Bini, David Bortz, Calin Chin-
dri§, Gabi Farca§, Milena Hering, Paul Horja, Bogdan Ion, Alex Kuronya, Nguyen
Minh, Evangelos Moroukos, Mircea Mustafa, Tom Nevins, Sam Payne, Mike Roth,
Janis Stipins and Alex Yong.
Special thanks are due to Sergey Fomin, Allen Knutson and Ravi Vakil who helped
me become a better mathematician.
Most of all I am indebted to my advisor, Professor William Fulton, who guided
and inspired me throughout the graduate school.
Personally, I would like to thank my parents and my parents in law, and especially
my wife, Stanca Ciupe, for being the way she is.
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TABLE OF CONTENTS
DEDICATION............................................................................................................................ ii
ACKNOWLEDGEMENTS.................................................................................................... iii
LIST OF FIGURES ................................................................................................................ vi
LIST OF APPENDICES ....................................................................................................... vii
CHAPTER
1. Introduction................................................................................................................... 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. Preliminaries................................................................................................................... 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 facts....................................................................................... 17
2.2.2 Equivariant Schubert calculus on G /P ............................................. 19
2.3 Quantum cohomology.......................................................................................... 23
2.4 Equivariant quantum cohomology of the homogeneouss paces......................... 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
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4. Positivity in the equivariant quantum Schubert calculus................................ 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 Grassmannian........................................................................................................ 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
APPENDICES................................................................................................................................. 94
BIBLIOGRAPHY........................................................................................................................... 116
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LIST OF FIGURES
Figure
5.1 Example of a partition: p = 3, m = 7, A = (4,2,1)....................................................... 60
5.2 Example of partitions A and A- ...................................................................................... 61
5.3 The set of vertical steps of a partition: p = 3, m = 7,7(A) = {1,4,6}...................... 71
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LIST OF A PPEN DICES
Appendix
A. Gysin maps .......................................................................................................................... 95
A.l The non-equivariant case.................................................................................... 95
A.2 Finite dimensional approximations and equivariant Gysin maps ......................102
B. Equivariant quantum cohomology - general definition and proofs of its properties . . 106
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CH A PTER 1
Introduction
1.1 Motivation
Inspired by physicists, a new deformation of the cohomology of a (smooth) variety
X, called (small) quantum cohomology, has been recently constructed. It encodes
certain enumerative information about the variety X. Understanding quantum co
homology in general proved to be a very difficult task and it is known only in some
limited cases (see e.g. [1] and references therein).
It has been better understood when X is a homogeneous space G/P, where G is
a complex, semisimple, connected Lie group and P a parabolic subgroup of G. In
this case the classical cohomology of X is an algebra with an additive basis consist
ing of Schubert classes, and the multiplicative structure constants are the famous
(generalized) Littlewood-Richardson coefficients, which play a fundamental role in
various ares of mathematics, such as algebraic geometry, algebraic combinatorics,
representation theory.
The quantum cohomology of A = G/P is an algebra which is a deformation of the
classical cohomology. It has an additive basis determined by the Schubert classes
of X, and its multiplicative structure constants, which generalize the Littlewood-
Richardson coefficients, are the (3-point, genus 0) Gromov-Witten invariants. These
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2
are positive integers equal to the number of rational curves of a certain degree pass
ing through three Schubert varieties in general position, whose dimensions add up
to the dimension of X. The associativity of quantum cohomology gives a systematic
approach for finding such numbers. In fact, since X is a homogeneous space, the
situation is more fortunate. The richness of the geometry and of combinatorial prop
erties of X = G/P have been used to obtain better algorithms for the computation
of the Gromov-Witten invariants, which are particularly efficient when G is of type
A (e.g. G = PGL{m)){5, 25, 19, 16, 28, 57, 73],
However, even in this case, the story is far from complete, as there is no proven
positive combinatorial formula for these coefficients. In fact, in spite of much recent
progress [6, 68, 13, 16, 21] the only case in which there is such a (conjectural) formula
is the type A Grassmannian ([16]).
The initial motivation for the project undertaken in this thesis was to understand
better the quantum cohomology of homogeneous spaces in general, and, in particular,
to find a positive formula for the Gromov-Witten invariants of the Grassmannian. In
order to do that, we have considered an equivariant version of quantum cohomology,
named equivariant quantum cohomology. It was introduced by A. Givental and B.
Kim [32] with the same purpose - to study the quantum cohomology of homogeneous
spaces.
It turned out however that the equivariant quantum cohomology has many inter
esting properties in its own right, therefore deserving a closer study. Besides this, one
also gains a better insight into the (non-equivariant) quantum cohomology. In this
respect, we obtain a new algorithm to compute the Gromov-Witten invariants. Its
main ingredient is a certain recurrence formula satisfied by the structure constants
of the equivariant quantum cohomology ring. This formula seems to play a key part
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