Proceedings of the Summer School on

Representation Theory of Algebras,

Finite and Reductive Groups

held at the

“Babeş–Bolyai” University, Cluj-Napoca September 15–25, 1997

“Babeş–Bolyai” University, Cluj-Napoca

FOREWORD

The present volume grew out of five courses given at the Summer School “Representation theory of algebras, finite and reductive groups” at the University of Cluj-Napoca, Romania, September 1997. It is intended to be an introduction to some aspects of the great diversity in representation theory. Each of the five chapters starts from scratch and leads to open conjectures and actual research; to keep this volume in a manageable size, though, proofs are frequently omitted, but each chapter ends with hints and references for further reading.

Chapter I lays the foundations about representations of finite groups, providing some of the standard results needed in subsequent chapters and also giving a picture of some of the most prominent open conjectures at present. Chapter II treats derived categories and equivalences - the relevance of this subject has been discovered by J. Rickard in the context of blocks with cyclic defect groups and has, during the last decade, significantly influenced the further development of modular representation theory. Chapter III leads from an ad hoc introduction of classical concepts in block theory to its systematic treatment as developed from the 1980’s on by L. Puig, and includes also a short account on stable equivalences and their use in block theory. Chapter IV brings group theory and algebraic geometry together, describing the rich theory of algebraic and reductive groups. Number theoretical aspects come into the subject in Chapter V, where the group rings of finite groups over the integers are studied, motivated by the fundamental question, to what extent the structure of a finite group is determined by its integral group ring.

*

This Summer School was made possible by the financial support from the Soros Foundation, the CNRS and the University of Cluj-Napoca who provided the accomodation. Special thanks also to the lecturers, who came to Cluj to give their courses - and last but not least to A. Marcus, whose organizational skills contributed greatly to turn this Summer School into an outstanding experience for all participants.

Markus Linckelmann, Paris, April 1998

CONTENTS

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv

Andrei Marcus I. Groups, algebras, representations 1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Complete local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Lifting idempotents. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. Self-injective and symmetric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6. Operations with kG-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7. Relative projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8. Vertices and sources. The defect group of a block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9. The Green correspondence and the Brauer correspondence . . . . . . . . . . . . . . . . . . . . 21 10. Clifford theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 11. Characters and numerical invariants of blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Thorsten Holm II. Derived categories, derived equivalences and representations 33

1. Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2. Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 3. Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4. Tilting theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48 5. Brauer tree algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6. Applications and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Markus Linckelmann III. Blocks of finite groups and stable equivalences 67

1. Blocks, defect groups and vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2. Group algebras and groups acting on algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3. Pointed groups and the Brauer construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4. Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5. Stable categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6. The trivial intersection case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Meinolf Geck IV. Finite reductive groups 101

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2. Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3. Some general properties of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4. Frobenius maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5. Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6. Deligne–Lusztig characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7. A guide to further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Wolfgang Kimmerle V. A course on integral group rings 127

1. The problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2. Integral versus other representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3. Descriptions of group rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4. Torsion units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5. Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

LIST OF PARTICIPANTS

Boboc, Crina – Bucharest University, Faculty of Mathematics, Str. Academiei nr. 14, RO-70109 Bucharest.

e-mail: crina@al.math.unibuc.ro

Both, Nicolae – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Breaz, Simion Sorin – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Călinescu, Corina – Bucharest University, Faculty of Mathematics, Str. Academiei nr. 14, RO-70109 Bucharest. Călugăreanu, Grigore – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca.

e-mail: calu@math.ubbcluj.ro

Chindriş, Călin Ioan – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Chiş-Buiga, Mihai – West University Timişoara, B-dul V. Pârvan nr 14, RO-1900 Timişoara.

e-mail: chis@tim1.uvt.ro

Chiş-Buiga, Codruţa – West University Timişoara, B-dul V. Pârvan nr 14, RO-1900 Timişoara. Ciubotaru, Dan-Mihai – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Crivei, Iuliu – Technical University Cluj, Department of Mathematics, Str. I. Bariţiu nr. 1, RO-3400 Cluj-Napoca. Crivei, Septimiu – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca.

e-mail: crivei@math.ubbcluj.ro

Dăuş, Leonard Iulian – Bucharest University, Faculty of Mathematics, Str. Academiei nr. 14, RO-70109 Bucharest. Deaconu, Laurenţiu – University of Piteşti, RO-0300 Piteşti. Deneş, Carol – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Asadurian, Eduard – University of Piteşti, RO-0300 Piteşti. Geck, Meinolf – Equipe des Groupes Finis, UFR de Mathématiques, Université Paris 7 Denis Diderot, 2 Place Jussieu, 75251 Paris Cedex 05, France.

e-mail: gech@math.jussieu.fr

Györfi, Zsolt – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Hanchevici, Teodor-Călin – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Holm, Thorsten – Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Postfach 4120, D-39016 Magdeburg, Germany.

e-mail: Thorsten.Holm@mathematik.uni-magdeburg.de

Horváth, Erzsébet – Technical University Budapest, Department of Mathematics, Budapest 1521, Hungary.

e-mail: he@math.bme.hu

Iancu, Ana Lăcrimioara – North University Baia-Mare, RO-4800 Baia-Mare.

e-mail: lacri@univer.ubm.ro

Laslo, Graţiela – ”Petru Maior” University Târgu-Mureş, RO-4300 Târgu-Mureş.

e-mail: glaslo@uttgm.ro

Kimmerle, Wolfgang – Universität Stuttgart, Mathematisches Institut B, Pfaffenwaldring 57 (Vaihingen), 70550 Stuttgart, Germany.

e-mail: kimmerle@machnix.mathematik.uni-stuttgart.de

Linckelmann, Markus – Equipe des Groupes Finis UFR de Mathématiques Université Paris 7 Denis Diderot 2 Place Jussieu 75251 Paris Cedex 05, France.

e-mail: linckelm@math.jussieu.fr

Madar, Adrian – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Marcus, Andrei – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca.

e-mail: marcus@math.ubbcluj.ro

Modoi, George Ciprian – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Panaite, Florin – Bucharest University, Faculty of Mathematics, Str. Academiei nr. 14, RO-70109 Bucharest.

e-mail: fpanaite@al.math.unibuc.ro, dstefan@stoilow.imar.ro

Prisecaru, Laurenţiu – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Purdea, Ioan – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Szántó, Csaba – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca. Szántó, Erzsébet – Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Lehrstuhl Algebra 07743 Jena, Germany.

e-mail: m6szer@rz.uni-jena.de

Szegedy, Balázs – “Eötvös Loránd” University, Faculty of Mathematics, Budapest, Hungary.

e-mail: szegedy@cs.elte.hu

Szőke, Magdolna – “Eötvös Loránd” University, Faculty of Mathematics, Budapest, Hungary.

e-mail: Magdolna.Szoke@Math.RWTH-Aachen.DE

Tóth, László – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca,

e-mail: ltoth@math.ubbcluj.ro

Voicu, Nicoleta – “Transilvania” University Braşov, Faculty of Sciences, Department of Mathematics, RO-2200 Braşov.