Blocks in Deligne's category Rep(St) [PhD thesis]

91 Pages·2010·1.386 MB·English

by  Jonathan Comes

BLOCKS IN DELIGNE'S CATEGORY Rep(St)
by
JONATHAN COMES
A DISSERTATION
Presentedto the Department ofMathematics
and the Graduate Schoolofthe Universityof Oregon
in partial fulfillment ofthe requirements
for the degree of
Doctor ofPhilosophy
June 2010
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University ofOregon Graduate School
Confirmation ofApproval and Acceptance ofDissertation prepared by:
Jonathan Comes
Title:
"Blocks inDeligne's Category Rep(S_t)"
This dissertationhas been accepted and approved inpartial fulfillment ofthe requirements for
the Doctor ofPhilosophy degree inthe Department ofMathematics by:
Victor Ostrik, Chairperson, Mathematics
Daniel Dugger, Member, Mathematics
Jonathan Brundan, Member, Mathematics
AlexanderKleshchev, Member, Mathematics
Michael Kellman, Outside Member, Chemistry
and Richard Linton, Vice Presidentfor Research and Graduate Studies/Dean ofthe Graduate
School for the University ofOregon.
June 14,2010
Original approval signatures are onfile withthe Graduate School and the UniversityofOregon
Libraries. '
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@201O, Jonathan Comes
--------
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An Abstract of the Dissertation of
Jonathan Comes for the degree of Doctor ofPhilosophy
in the Department ofMathematics to be taken June 2010
Title: BLOCKS IN DELIGNE'S CATEGORY Rep(St)
Approved: _
Dr. Victor Ostrik
We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an
integer. Thereafter, we give a completedescriptionofthe blocks in Rep(St) for arbitraryt. Finally,
we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St).
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CURRICULUM VITAE
NAME OF AUTHOR: Jonathan Comes
PLACE OF BIRTH: Great Falls, MT, U.S.A.
DATE OF BIRTH: August 12, 1981
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University ofOregon, Eugene, OR
University ofMontana, Missoula, MT
DEGREES AWARDED:
Doctor ofPhilosophy in Mathematics, University ofOregon, 2010
Master ofArts in Mathematics, University ofMontana, 2004
Bachelor ofArts in Mathematics, University ofMontana, 2003
AREAS OF SPECIAL INTEREST:
Tensor Categories
Combinatorial Representation Theory
Diagram Algebras
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University ofOregon, 2004- 2010
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ACKNOWLEDGMENTS
First and foremost I would like to thank my advisor Victor Ostrik. His guidance and
patience have been vital throughout the last five years. Thank you so very much Victor. Next,
I want to thank Sasha Kleshchev for suggesting this problem. I would also like to thank Jon
Brundanfor providingme with an excitingintroductionto representation theory. I am grateful to
both Jon and Sasha for numerous valuable conversations.
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TABLE OF CONTENTS
Chapter Page
INTRODUCTION
II PRELIMINARIES .... 3
II.1 Tensor Categories 3
II.2 Linear Algebra in Tensor Categories . 6
II.3 Blocks and Semisimple Categories .. 7
11.4 Pseudo-abelian Envelopes. . . . . . . 8
II.5 Pseudo-abelianEnvelopesofTensor Categories . 11
11.6 Notation and Conventions for Young Diagrams. 12
III THE TENSOR CATEGORY Rep(St;F) . 14
III.1 Motivation: Representations ofSd and PartitionDiagrams 14
III.2 Definition ofRep(St;F) . 21
III.3 The Trace ofan Endomorphism in &Po(St; F) . 25
IV INDECOMPOSABLE OBJECTS . 27
IV.1 Classification ofIndecomposable Objects in Rep(St;F) 27
IV.2 On Rep(St;F) for Generic t . 33
IV.3 The Interpolation Functor &P(Sd;F) --t Rep(Sd;F) . 35
IVA Dimensions . . . . . . . . . . . . . . . . . . . . . 38
V ENDOMORPHISMS OF THE IDENTITY FUNCTOR 41
V.1 Interpolating Sums ofr-cycles 41
V.2 Frobenius' Formula . 44
VI BLOCKS OF INDECOMPOSABLE OBJECTS 46
VI.1 What Does Frobenius' FormulaTell Us About Blocks? 46
VI.2 On the Equivalence Relation ,.s . 47
VI.3 The Functor - ® £(0) 51
VIA Lift ofIdempotents . . . . . . . 54
VII QUIVER DESCRIPTION OF A NON-SEMISIMPLE BLOCK 59
VII.1 The Nontrivial Block in&P(So;F) . 59
VII.2 Comparison ofNon-semisimple Blocks in&P(Sd;F) 62
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Chapter Page
VII.3 Description ofBlocks Via Martin . 63
VIII DECOMPOSING TENSOR PRODUCTS. 66
VIII.1 The Generic Case . . . . . 67
VIII.2 The Non-semisimple Case. 71
IX TENSOR IDEALS . . . . 72
IX.1 Deligne's Lemma. 72
IX.2 ProofofTheorem IX.0.2 73
APPENDIX: LIST OF SYMBOLS . . 76
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 80
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LIST OF FIGURES
Figure Page
Chapter II
1 A Young Diagram 13
Chapter III
1 A Partition Diagram . . . . . . . . . 15
2 Composition ofPartition Diagrams. 19
3 The Hexagon Axiom . . . . . . . . . 24
4 The Rigidity Axioms. . . . . . . . . 24
5 The Trace of'if (Left) and the Trace Diagram of'if (Right) . 25
6 An Example ofa Trace Diagram . . . . . . . . . . . . . . . 26
Chapter IV
1 An Example ofHook Lengths . 38
2 The (.\,d) Grid Marking ... 39
Chapter V
1 A Perfect (i,i')-coloringof'if 42
Chapter VI
1 Decomposing £(2,1,0,...)0 £(0).. 52
2 Constructing p(O) From .\(0) ..... 54
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LIST OF TABLES
Table Page
Chapter VIII
1 Nonzero r~,61) for Various>. . 67
2 Calculations for Computing (12) ®(2,1) 68
3 Calculations for Computing (12) ®(22) . 68
4 Calculations for Computing (2,1)®(2,1) 69
5 Calculations for Computing (2,1) ®(22) 70