REPRESENTATIONS OF KHOVANOV-LAUDA-ROUQUIER ALGEBRAS OF AFFINE LIE TYPE by ...

314 Pages·2016·1.18 MB·English

REPRESENTATIONS OF KHOVANOV-LAUDA-ROUQUIER ALGEBRAS OF
AFFINE LIE TYPE
by
ROBERT W. MUTH
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2016
DISSERTATION APPROVAL PAGE
Student: Robert W. Muth
Title: Representations of Khovanov-Lauda-Rouquier Algebras of Affine Lie Type
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Alexander Kleshchev Chair
Jonathan Brundan Co-chair
Arkady Berenstein Core Member
Victor Ostrik Core Member
Michael Kellman Institutional Representative
and
Scott Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2016
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(cid:13)c 2016 Robert W. Muth
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DISSERTATION ABSTRACT
Robert W. Muth
Doctor of Philosophy
Department of Mathematics
June 2016
Title: Representations of Khovanov-Lauda-Rouquier Algebras of Affine Lie Type
We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine
Lie type. Associated to every convex preorder on the set of positive roots is a
system of cuspidal modules for the KLR algebra. For a balanced order, we study
imaginary semicuspidal modules by means of ‘imaginary Schur-Weyl duality’. We
then generalize this theory from balanced to arbitrary convex preorders for affine
ADEtypes. Undertheassumptionthatthecharacteristicofthegroundfieldisgreater
than some explicit bound, we prove that KLR algebras are properly stratified. We
introduceaffinezigzagalgebrasandprovethattheseareMoritaequivalenttoarbitrary
imaginary strata if the characteristic of the ground field is greater than the bound
mentioned above. Finally, working in finite or affine affine type A, we show that skew
Specht modules may be defined over the KLR algebra, and real cuspidal modules
associated to a balanced convex preorder are skew Specht modules for certain explicit
hook shapes.
This dissertation contains previously published (unpublished) co-authored
material.
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CURRICULUM VITAE
NAME OF AUTHOR: Robert W. Muth
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
University of Arizona, Tucson, AZ
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2016, University of Oregon
Bachelor of Science, Mathematics, 2011, University of Arizona
Bachelor of Fine Arts, Visual Arts, 2004, University of Arizona
AREAS OF SPECIAL INTEREST:
Representation Theory
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Mathematics, University of Oregon,
Eugene OR, 2011–2016
PUBLICATIONS:
Imaginary Schur-Weyl duality (joint with A. Kleshchev), Mem. Amer. Math.
Soc. (to appear), 82 pages
Graded skew Specht modules and cuspidal modules for Khovanov-Lauda-Rouquier
algebras of affine type A, 32 pages (submitted)
Stratifying KLR algebras of affine Lie type (joint with A. Kleshchev), 25 pages
(submitted)
Affine zigzag algebras and imaginary strata for KLR algebras (joint with A.
Kleshchev), 33 pages (submitted)
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ACKNOWLEDGEMENTS
First and foremost, I must thank my advisor, Alexander Kleshchev, for his
patient guidance, insight, and support over the past four years. I have also learned
much from Jon Brundan, Victor Ostrik, and Arkady Berenstein. Klaus Lux first
introduced me to the beauty of representation theory as an undergraduate, and Pham
Huu Tiep pointed me toward Oregon for graduate school, and for that I am grateful.
Thank you to my friends and family for their support and encouragement, my
fellow graduate students for their camaraderie and many enlightening conversations,
and the math department office staff for all their kind assistance. Thank you to my
kids, Otto and Loretta, for putting up with all the times I was doing math when I
should have been playing hide-and-seek. Most of all, thank you to my wife Lindsey—
without your support and sacrifice this grad school adventure would never have been
possible.
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For Lindsey
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Cuspidal systems . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Imaginary Schur-Weyl duality . . . . . . . . . . . . . . . . . . 6
1.3. Stratifying KLR algebras of affine ADE types . . . . . . . . . 12
1.4. Skew Specht modules and real cuspidal modules . . . . . . . . 15
1.5. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1. Ground rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2. Symmetric groups and Schur algebras . . . . . . . . . . . . . . 18
2.3. Lie theoretic notation . . . . . . . . . . . . . . . . . . . . . . . 29
2.4. KLR algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5. Convex preorders and root partitions . . . . . . . . . . . . . . 44
2.6. Cuspidal systems and standard modules . . . . . . . . . . . . . 45
III. IMAGINARY SCHUR-WEYL DUALITY . . . . . . . . . . . . . . . . 47
3.1. Imaginary tensor space . . . . . . . . . . . . . . . . . . . . . . 47
3.2. Imaginary Schur-Weyl duality . . . . . . . . . . . . . . . . . . 56
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Chapter Page
3.3. Imaginary Schur algebras . . . . . . . . . . . . . . . . . . . . . 61
3.4. Imaginary induction and restriction . . . . . . . . . . . . . . . 64
3.5. Imaginary Howe duality . . . . . . . . . . . . . . . . . . . . . . 67
3.6. Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . 88
3.7. Alternative definitions of standard modules . . . . . . . . . . . 97
3.8. Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.9. Ringel duality and double centralizer properties . . . . . . . . 104
3.10. Characters of imaginary modules . . . . . . . . . . . . . . . . 108
IV. STRATIFYING KLR ALGEBRAS OF SYMMETRIC AFFINE LIE
TYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.1. Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2. Semicuspidal modules . . . . . . . . . . . . . . . . . . . . . . . 128
4.3. Stratifying KLR algebras . . . . . . . . . . . . . . . . . . . . . 137
4.4. Reduction modulo p of irreducible and standard modules . . . 149
4.5. Zigzag algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.6. The minuscule imaginary stratum category . . . . . . . . . . . 166
4.7. On the higher imaginary stratum categories . . . . . . . . . . 179
V. SKEW SPECHT MODULES AND REAL CUSPIDAL MODULES IN
TYPE A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.2. Manipulating elements of KLR algebras . . . . . . . . . . . . . 195
5.3. Skew Specht modules . . . . . . . . . . . . . . . . . . . . . . . 197
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Chapter Page
5.4. Restrictions of Specht modules . . . . . . . . . . . . . . . . . . 203
5.5. Joinable diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 222
5.6. Cuspidal systems . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.7. Cuspidal modules and skew hook Specht modules . . . . . . . 234
VI. CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.1. Imaginary tensor space for non-simply-laced types . . . . . . . 244
6.2. Proofs of zigzag relations . . . . . . . . . . . . . . . . . . . . . 283
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
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