Topology of the Affine Springer Fiber in Type A

84 Pages·2017·0.64 MB·English

by  Tobias Wilson

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Doctoral Dissertations Dissertations and Theses
March 2016
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Tobias Wilson
University of Massachusetts Amherst
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Wilson, Tobias, "Topology of the Affine Springer Fiber in Type A" (2016). Doctoral Dissertations. 610.
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TOPOLOGYOFTHEAFFINESPRINGERFIBERINTYPEA
ADissertationPresented
by
TOBIASWILSON
SubmittedtotheGraduateSchoolofthe
UniversityofMassachusettsAmherstinpartialfulfillment
oftherequirementsforthedegreeof
DOCTOROFPHILOSOPHY
February2016
DepartmentofMathematicsandStatistics
(cid:13)c CopyrightbyTobiasWilson2016
AllRightsReserved
TOPOLOGYOFTHEAFFINESPRINGERFIBERINTYPEA
ADissertationPresented
by
TOBIASWILSON
Approvedastostyleandcontentby:
AlexeiOblomkov,Chair
TomBraden,Member
JuliannaTymoczko,Member
AndrewMcGregor
ComputerScience,OutsideMember
FarshidHajir,DepartmentHead
MathematicsandStatistics
ACKNOWLEDGEMENTS
I would like to acknowledge my indebtedness to my advisor, Alexei Oblomkov, for
his patience, guidance, and advice throughout the past 5 years. His assistance, sug-
gestions, explanations, and re-explanations made this work possible. I am extremely
grateful to Tom Braden, Julianna Tymoczko, and Andrew McGregor for many helpful
conversationsaboutmywork.
I have received a very great deal of support from family and friends while in grad-
uate school. Fellow mathematicians Nico, Luke, Jeff, Steve, Jenn, and Tom provided
much needed commiseration, advice, and occasional distraction. Laura and Nicky, al-
ways integral parts of my educational career, offered regular sanity checks. Cat and
Rosie have given me near-daily encouragement and, towards the end, a second home.
My parents have been a constant source of support and advice. I am indebted to all of
them,andothers.
Andfinally,IoweendlessthankstoAndy,whohasbeenmyfavoritepartofthelast
6years.
iv
ABSTRACT
TOPOLOGYOFTHEAFFINESPRINGERFIBERINTYPEA
FEBRUARY2016
TOBIASWILSON
M.S.,UNIVERSITYOFMASSACHUSETTSAMHERST
Ph.D.,UNIVERSITYOFMASSACHUSETTSAMHERST
Directedby: ProfessorAlexeiOblomkov
We develop algorithms for describing elements of the affine Springer fiber in type
A for certain γ ∈ g(C[[t]]). For these γ, which are equivalued, integral, and regular,
it is known that the affine Springer fiber, X , has a paving by affines resulting from
γ
the intersection of Schubert cells with X . Our description of the elements of X allow
γ γ
us to understand these affine spaces and write down explicit dimension formulae. We
also explore some closure relations between the affine spaces and begin to describe the
momentmapfortheboththeregularandextendedtorusaction.
v
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 AffineSpringerFibers . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 MainResultsandOrganization . . . . . . . . . . . . . . . . . . . . 2
2. AFFINEGRASSMANNIANSANDAFFINESPRINGERFIBERS . . . . . 3
2.1 TheAffineGrassmannian . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3 TorusAction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 AffineSpringerFiber . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 EquivariantCohomology . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 EquivariantCohomologyfortheAffineSpringerFiber . . . . . . . 9
3. DESCRIPTIONOFTHEAFFINESPACES . . . . . . . . . . . . . . . . . . 10
3.1 LatticeNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Two-DimensionalCase . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Three-DimensionalCase . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 GeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 LatticesandSchubertCells . . . . . . . . . . . . . . . . . . . . . . . 22
4. CLOSURERELATIONSHIPS . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 TwoDimensionalCase . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.1 LatticeDecompositions . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.3 FormingK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.4 ClosurePicture . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 ThreeDimensionalCase . . . . . . . . . . . . . . . . . . . . . . . . 30
vi
4.3.1 ConstructingLatticesFromGr(K,m)t,γ . . . . . . . . . . . . 30
4.3.2 SummarizingLattices . . . . . . . . . . . . . . . . . . . . . . 34
4.4 SummarizingDimensionsandClosureRelations . . . . . . . . . . 35
4.5 GeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5. ONE-DIMENSIONALORBITSANDMOMENTGRAPHS . . . . . . . . 42
5.1 ZeroandOne-DimensionalOrbits . . . . . . . . . . . . . . . . . . 42
5.2 MomentGraphforn = 2andn = 3 . . . . . . . . . . . . . . . . . . 44
5.2.1 Index0Lattices . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.2 Index1and2Lattices . . . . . . . . . . . . . . . . . . . . . . 48
5.2.3 One-DimensionalOrbits . . . . . . . . . . . . . . . . . . . . 51
6. DIRECTIONSFORFUTUREWORK . . . . . . . . . . . . . . . . . . . . . 54
APPENDICES
A. POSSIBLELATTICETYPESWHENn = 3 . . . . . . . . . . . . . . . . . . 56
B. CLOSUREIN3DIMENSIONALCASE . . . . . . . . . . . . . . . . . . . 66
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
vii
LIST OF TABLES
Table Page
1. LatticeSubsetsInGenericClosure . . . . . . . . . . . . . . . . . . . . . . . 37
viii
LIST OF FIGURES
Figure Page
1. Two-dimensionallatticesandclosures . . . . . . . . . . . . . . . . . . . . . 30
2. Sub-varietiesintheclosureofagenericlatticevariety . . . . . . . . . . . . 37
3. Verticesoflatticeswithindex0labeledbyminimumdegree . . . . . . . . . 45
4. Arrangementofthevertices,coloredbytype. . . . . . . . . . . . . . . . . . 46
5. Arrangementofthevertices,coloredbydimension. . . . . . . . . . . . . . 47
6. Index1lattices,withverticescoloredbytype. . . . . . . . . . . . . . . . . . 49
7. Index1lattices,withverticescoloredbydimension. . . . . . . . . . . . . . 49
8. Index2vertices,coloredbytype. . . . . . . . . . . . . . . . . . . . . . . . . 50
9. Index2vertices,coloredbydimension. . . . . . . . . . . . . . . . . . . . . 50
10. One-dimensionalorbitsforwidelyspaceddegreetuples . . . . . . . . . . . 52
11. Allone-dimensionalorbitsinindex0 . . . . . . . . . . . . . . . . . . . . . . 53
ix