CONTEMPORARYMATHEMATICS

American Mathematical Society

478

Representation TheoryFourth International Conference

on Representation Theory July 16 –20, 2007

Lhasa, China

Zongzhu Lin Jianpan Wang

Editors

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American Mathematical SocietyProvidence, Rhode Island

CONTEMPORARYMATHEMATICS

478

Representation Theory

Fourth International Conference on Representation Theory

July 16–20, 2007 Lhasa, China

Zongzhu Lin Jianpan Wang

Editors

Editorial Board

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classification. Primary 16Gxx, 17Bxx, 20Cxx, 20Gxx;Secondary 17B10, 17B20, 17B37, 17B45, 17B56, 20G05, 20G10, 20G42, 20C05, 20C08,

20C30.

Library of Congress Cataloging-in-Publication Data

Representation theory / Zongzhu Lin, Jianpan Wang, editors.p. cm. — (Contemporary mathematics ; v. 478)

Includes bibliographical references.ISBN 978-0-8218-4555-4 (alk. paper)1. Representations of algebras—Congresses. 2. Representations of groups—Congresses.

I. Lin, Zongzhu. II. Wang, Jianpan, 1949– III. Title.

QA150 .R46 2008515′.7223—dc22 2008034291

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Contents

Preface vii

List of Talks ix

List of Participants xiii

Sum formulas and Ext-groupsHenning Haahr Andersen 1

Schur-Weyl duality in positive characteristicStephen Doty 15

The centers of Iwahori-Hecke algebras are filteredAndrew Francis and Weiqiang Wang 29

On Kostant’s theorem for Lie algebra cohomologyUniversity of Georgia VIGRE Algebra Group 39

G-stable pieces and partial flag varietiesXuhua He 61

Steinberg representations and duality properties of arithmetic groups, mappingclass groups, and outer automorphism groups of free groups

Lizhen Ji 71

Characters of simplylaced nonconnected groups versus characters ofnonsimplylaced connected groups

Shrawan Kumar, George Lusztig, and Dipendra Prasad 99

Classification of finite-dimensional basic Hopf algebras according to theirrepresentation type

Gongxiang Liu 103

Twelve bridges from a reductive group to its Langlands dualG. Lusztig 125

Some new highest weight categoriesBrian J. Parshall and Leonard L. Scott 145

Classification of quasi-trigonometric solutions of the classical Yang–Baxterequation

Iulia Pop and Alexander Stolin 155

The relevance and the ubiquity of Prüfer modulesClaus Michael Ringel 163

v

vi CONTENTS

Quivers and the euclidean groupAlistair Savage 177

eu2-Lie admissible algebras and Steinberg unitary Lie algebrasShikui Shang and Yun Gao 189

Lusztig’s conjecture for finite classical groups with even characteristicToshiaki Shoji 207

A survey on quasifinite representations of Weyl type Lie algebrasYucai Su 237

Maximal and primitive elements in baby Verma modules for type B2Nanhua Xi 257

Irreducible representations of the special algebras in prime characteristicYu-feng Yao and Bin Shu 273

Preface

The fourth International Conference on Representation Theory (ICRT-IV) was held in Lhasa, China, during July 16–20, 2007. The first three conferenceswere held in Shanghai (1998), Kunming (2001), and Chengdu (2004). The main goalof the ICRT is to bring together representation theorists of various subjects suchas algebraic groups, quantum groups, finite groups, Lie algebras, vertex operatoralgebras, Hecke algebras and complex reflection groups, quivers, finite dimensionalalgebras, Hall algebras, and other related topics. The representation theory hasevolved to connect many different fields of mathematics and the ICRT intends tocommunicate the common ideas and approaches to common questions. Duringthe five day programs, participants have endured both mathematical and physicalchallenges. There was also a two day sub-conference on Mathematical Historywhich covered many interesting historical topics such as early mathematics textbooks in Tibet and the first Chinese national who ever obtained formal Ph.D. inmathematics.

This volume contains eighteen papers that more or less reflect the topics of theconference. All papers were carefully refereed. Several referees read several versionsof the manuscripts and made critical comments to ensure the high quality of thepapers. The editors want to express their sincere appreciation to all referees fortheir anonymous contribution. Due to page limitation, the editors regret that notall submitted papers are included and thank all authors for their contribution tothis proceedings.

It was a challenge from many respects to bring the conference to this ”highlevel”. The conference would not be possible without many people’s hardwork.Jianpan Wang and Hebing Rui had traveled to Lhasa several times to arrange hoteland lecture facilities. Both of them have been working for more than a year raisingfund for the conference and making travel arrangements for international partic-ipants. Many participants took trains through the newly built railroad at 5 kmaltitude and the train ride lasts for more than two days. Making such arrangementwas impossible without the help of many local mathematicians at various cities. Themathematicians in Tibet University, Da Luosang Langjie and Yutian Fei, providedinvaluable help making local arrangement including the post-conference tours. Thevolunteer staff member, Xin Wang (a graduate student at University of Virginia),provided an impeccable professional-type services during conference.

The financial support from the following organizations, projects, and individu-als are greatly appreciated:

— Tianyuan Mathematical Fund, National Natural Science Foundation ofChina

— “985” fund, East China Normal University

vii

viii PREFACE

— “111 Project”, Department of Mathematics, East China Normal Univer-sity

— Changjiang Scholars and Innovative Research Team, Department of Math-ematics, East China Normal University

— Jun Hu, “New Century Excellent Talents in University”, Beijing Instituteof Technology

— Fang Li, “New Century Excellent Talents in University”, Zhejiang Uni-versity

— Hebing Rui, “New Century Excellent Talents in University”, East ChinaNormal University

— Yucai Su, “One-Hundred Talent Program”, University of Science andTechnology of China

— Jianpan Wang, Fund of Science and Technology Commission of ShanghaiMunicipality, No. 06JC 14024

— Jianpan Wang, fund of National Natural Science Foundation of China,No. 10631010

The ICRT-IV was co-hosted by East China Normal University and the TibetUniversity at Lhasa. The organizing committee consists of Jianpan Wang (Chair,East China Normal University), Da Luosang Langjie (Co-chair, Tibet Universityat Lhasa), Bangming Deng (Beijing Normal University), Yutian Fei (Tibet Uni-versity), Jun Hu (Beijing Institute of Technology), Ruyun Ma (Northwest NormalUniversity), Liangang Peng (Sichuan University), Hebing Rui (East China NormalUniversity).

The program committee consists of Jianpan Wang (Chair, East China Nor-mal University), Jiping Zhang (Co-chair, Peking University), Chongying Dong(UC Santa Cruz), Jie Du (University of New South Wales), Zongzhu Lin (KansasState University), Yucai Su (The University of Science and Technology of Chi-nese), Weiqiang Wang (University of Virginia), Nanhua Xi (Academy Sinica), JieXiao (Tsinghua University).

Last but not least, the editors want to thank Christine M. Thivierge, AssociateEditor for Proceedings at American Mathematical Society who worked so closelyand patiently with the editors to ensure the timely publication of this proceedings.

Zongzhu LinJianpan Wang

June 2008

List of Talks

Principal Speakers

Henning H. AndersenSum formulas and Ext-groups

Susumu ArikiCrystal theory and hecke algebras of type Bn

Michel BrouéFamilies of unipotent characters and cyclotomic algebras

Jonathan BrundanCyclotomic Hecke algebras and parabolic category O

Jon CarlsonModules of constant Jordan type

Shun-Jen ChengKostants homology formula for infinite-dimensional lie superalgebras

Chongying DongRepresentation theory for vertex operator algebras

Yun GaoIrreducible Wakimoto-like modules for lie algebras of type A

Shrawan KumarSpecial isogenies and tensor product multiplicities

George LusztigUnipotent elements in small characteristic

Brian ParshallSome results on quantum and algebraic group cohomology and applications

Liangang PengConical extensions of derived categories

Claus M. RingelThe relevance and the ubiquity of prüfer modules

Olivier SchiffmannGeometric construction of macdonald polynomials

Leonard ScottSome Z/2-graded representation theory

Toshiaki Shoji,Lusztigs conjecture for finite classical groups

Yucai SuQuasifinite representations of some lie algebras containing the Virasoroalgebra

ix

x LIST OF TALKS

Toshiyuki TanisakiDifferential operators on quantized flag manifolds at roots of 1

Nanhau XiSome maximal elements in a baby Verma module

Contributed 30 Minute Talks

Thomas BrüstleGentle algebras given by surface triangulations

Vlastimil DlabStandardly stratified approximations

Jie DuQuantum gl∞, infinite q-Schur algebras and their representations

Ming FangDominant dimensions and double centralizer properties

Andrew FrancisSymmetric polynomials of Jucys-Murphy elements and the centre of theIwahori-Hecke algebra of type A

Xuhua HeMinimal length elements and G-stable pieces

Jun HuMorita equivalences of cyclotomic hecke algebras of type G(r, p, n)

Lizhen JiDuality of arithmetic groups, mapping class groups and outer automor-phisms groups of free groups

Otto KernerCluster tilted algebras of rank three

Yanan Lin,Generic sheaves over elliptic curves

Zongzhu LinCanonical bases for algebras arising from quivers with loops

Gongxiang LiuClassification of finite dimensional basic hopf algebras according to theirrepresentation type

Alistair SavageA geometric construction of a crystal graph commutor

Bin ShuRepresentations and cohomology of Jacobson-Witt algebras

Mei Si,Discriminants of Brauer algebras

Alexander StolinClassification of quantum groups

Liping WangLeading coefficients of the Kazhdan-Lusztig polynomials of the affine Weylgroup of type B̃2

Adrian Leonard WilliamsMinimal dimensions of some irreducible representations of the symmetricgroup

Jiping ZhangThe block union of finite groups

LIST OF TALKS xi

Mathematical History TalksTom Archibald

Chinese mathematicians and the international research community: thecase of Hu Mingfu and integral equations

Elena AusejoCommercial arithmetic in the Spanish renaissance

Jose A. CerveraThe Chou Suan by Giacomo Rho: An example of mathematical adaptationin China

Joseph W. DaubenZhu Shijie and the Jade Mirror of the Four Unknowns

Qi HanAntoine Thomas (1644C1709) and the first introduction of western algebrainto china

Da Luosang LangjieDuchung Zurtsi : A mathematics textbook for secular and official schoolsof Tibet

Karen H. Parshall4000 Years of algebra: an historical tour from BM 13901 to ModerneAlgebra

Yongdong PengThe early communication of cybernetics in China (1929C1966)

Yibao XuChinese gougu theory versus Euclidean geometry: views of a seventeenth-century Chinese mathematician

David E. ZitarelliMiss Mullikin and the internationalization of topology

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List of Participants

Henning H. AndersenAarhus University, DENMARK

Tom ArchibaldSimon Fraser University, CANADA

Susumu ArikiKyoto University, JAPAN

Elena AusejoUniversity of Zaragoza, SPAIN

Xiaotang BaiNankai University, Tianjing

Stephen BermanUniversity of Saskatchewan, CANADA

Michel BrouéInstitut Henri-Poincaré Paris,FRANCE

Jonathan BrundanUniversity of Oregon, USA

Thomas BrüstleUniversité de Sherbrooke, CANADA

Jon F. CarlsonUniversity of Georgia, USA

Bintao CaoInstitute of Mathematics, AcademiaSinica, Beijing

Jose A. CerveraITESM, Campus Monterrey, MEXICO

Shun-Jen ChengAcademia Sinica (Taipei), Taipei

Joseph W. Dauben,The City University of New York, USA

Bangming DengBeijing Normal University, Beijing

Vlastimil DlabCarleton University, CANADA

Chongying DongUniversity of California at Santa Cruz,USA

Jie DuUniversity of New South Wales,AUSTRALIA

Zhoutian FanBeijing Normal University, Beijing

Ming FangBeijing University of Technology,Beijing

Yutian FeiTibet University, Lhasa

Andrew FrancisUniversity of West Sydney,AUSTRALIA

Qiang FuTongji University, Shanghai

Yun GaoYork University, CANADA

Jingyun GuoXiangtan University, Xiangtan

Qi HanInstitute for History of Natural Science,Academia Sinica, Beijing

Yang HanInstitute of Systems Science, AcademiaSinica, Beijing

xiii

xiv LIST OF PARTICIPANTS

Xuhua HeStony Brook University, USA

Terrell L. HodgeWestern Michigan University, USA

Masaharu KanedaOsaka City University, JAPAN

Shrawan KumarUniversity of North Carolina, USA

Jiuzu HongInstitute of Mathematics, AcademiaSinica, Beijing

Jun HuBeijing Institute of Technology, Beijing

Lizhen JiUniversity of Michigan, USA

Otto T. KernerHeinrich-Heine-Universität,GERMANY

Jonathan KujawaUniversity of Georgia, USA

Da Luosang LangjieTibet University, Lhasa

Fang LiZhejiang University, Hangzhou

Lei LinEast China Normal University,Shanghai

Yanan LinXiamen University, Xiamen

Zongzhu LinKansas State University, USA

Gongxiang LiuInstitute of Mathematics, AcademiaSinica, Beijing

Li LuoInstitute of Mathematics, AcademiaSinica, Beijing

George LusztigMIT, USA

Frantǐsek MarkoPenn State University, Hazleton, USA

Qingnian PanHuizhou University, Huizhou

Brian J. ParshallUniversity of Virginia, USA

Karen H. ParshallUniversity of Virginia, USA

Lianggang PengSichuan University, Chengdu

Yongdong PengHubei Education Press, Wuhan

Claus M. RingelUniversity of Bielefeld, GERMANY

Hebing RuiEast China Normal University,Shanghai

Alistair SavageUniversity of Ottawa, CANADA

Olivier SchiffmannÉcole Normale Supérieure, FRANCE

Leonard ScottUniversity of Virginia, USA

Jianyi ShiEast China Normal University,Shanghai

Toshiaki ShojiNagoya University, JAPAN

Bin ShuEast China Normal University,Shanghai

Mei SiEast China Normal University,Shanghai

Alexander StolinUniversity of Gothenburg, SWEDEN

Yucai SuUniversity of Science and Technology ofChina, Hefei

LIST OF PARTICIPANTS xv

Toshiyuki TanisakiOsaka City University, JAPAN

Jianpan WangEast China Normal University,Shanghai

Li WangShanghai Normal University, Shanghai

Liping WangInstitute of Mathematics, AcademiaSinica, Beijing

Weiqiang WangUniversity of Virginia, USA

Xiaoming WangEast China Normal University,Shanghai

Xin WangUniversity of Virginia, USA

Adrian WilliamsImpare College, University of London,UK

Nanhua XiAcademia Sinica, Beijing

Yibao XuThe City University of New York, USA

Rong YanInstitute of Mathematics, AcademiaSinica, Beijing

Ziting ZengBeijing Normal University, Beijing

Jiping ZhangPeking University, Beijing

Pu ZhangShanghai Jiaotong University, Shanghai

Qinhai ZhangShanxi Normal University, Xi’an

Shizhuo ZhangBeijing University of Technology,Beijing

David E. Zitarelli,Temple University, USA

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Articles in this volume cover topics related to representation theory of various algebraic objects such as algebraic groups, quantum groups, Lie algebras, (finite- and infinite-dimen-sional) finite groups, and quivers. Collected in one book, these articles show deep relations between all these aspects of Representation Theory, as well as the diversity of algebraic, geometric, topological, and categorical techniques used in studying representations.

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