Progress in Mathematics Volume 95

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2 Progress in Mathematics Volume 95 Series Editors J. Oesterle A. Weinstein

3 Representation Theory of Finite Groups and Finite-Dimensional Algebras Proceedings of the Conference at the University of Bielefeld from May 15-17,1991, and 7 Survey Articles on Topics of Representation Theory Edited by G. o. Michler C.M. Ringel Springer Basel AG

4 Editor's addresses: Prof. Dr. G. O. Michler Institut flir Experimentelle Mathematik Universitiit GHS Essen Ellernstrasse Essen 12 Prof. Dr. C. M. Ringel Universitiit Bielefeld Fakultiit flir Mathematik Universitiitsstrasse Bielefeld library of Congress Cataloging-in-Pubtication Data Representation tbeory of finite groups and finite-dimensional algebras / edited by G. O. Michler, C. M. Ringel. p. em. - (Progress in mathematics ; v. 95) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Finite groups. 2. Representations of groups. 3. Representations of algebras. I. Michler, G. (Gerhard), II. Ringel, Claus Michael. III. Title: Finite-dimensional algebras. IV. Series: Progress in mathematics (Boston, Mass.) vol. 95 QA171.R '.2 - dc20 Deutsche Bibtiothek Cataloging-in-Pobtication Data Representation tbeory of finite groups and finite-dimensional algebras / ed. by G. O. Michler; C. M. Ringel. - Basel Boston ; Berlin : Birkhiiuser, 1991 (Progress in mathematics ; Vol. 95) ISBN NE: Michler, Gerhard O. [Hrsg.]; GT This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to»verwertungsgesellschaft Wort«, Munich Springer Basel AG Originally published by Birkhluser Verlag Basel in 1991 Softcover reprint of the hardcover 1st edition 1991 ISBN

5 v CONTENTS Preface Contributors vii Vlll 1. Survey Articles: B. Fischer B. Huppert K. Lux, H. Pahlings B.H. Matzat G.O. Michler C.M. Ringel K.W. Roggenkamp Clifford-Matrices Research in Representation Theory at Mainz ( ) Computational Aspects of Representation Theory Der Kenntnisstand in der konstruktiven Galoisschen Theorie Contributions to Modular Representation Theory of Finite Groups Recent Advances in the Representation Theory of Finite Dimensional Algebras The isomorphism problem for integral group rings of finite groups Research Articles: M. Auslander, I. Reiten Cohen-Macaulay and Gorenstein Artin Algebras J.A. Green M. Isaacs J.C. Jantzen Classical Invariants and the General Linear Group Partial characters of 7r-separable groups First Cohomology Groups for Classical Lie Algebras

6 vi C. Bessenrodt Endotrivial modules and the 317 Auslander-Reiten quiver E. Dieterich Tame curve singularities 327 with large conductor R. Dipper Polynomial representations of finite general 343 linear groups in non-describing characteristic K. Erdmann On Auslander-Reiten components 371 for wild blocks D. Happel On Gorenstein Algebras 389 G. Hiss Decomposition numbers of finite groups 405 of Lie type in non-defining characteristic R. Knorr Counting blocks of defect zero 419 B. Kiilshammer Group-theoretical descriptions of ring- 425 theoretical invariants of group algebras G. Malle Darstellungstheoretische Methoden 443 bei der Realisierung einfacher Gruppen vom Lie Typ als Galoisgruppen O. Manz Some new developments and open questions 461 in the character theory of finite groups W. Plesken Some applications of representation theory 477 A. Wiedemann A construction of orders 497 of finite global dimensions W. Willems Duality and forms in representation theory 509

7 vii PREFACE From April 1, 1984 until March 31, 1991 the Deutsche Forschungsgemeinschaft has sponsored the project "Representation Theory of Finite Groups and Finite Dimensional Algebras". The proposal for this project was submitted by B. Huppert (Mainz), B. Fischer (Bielefeld), G. Michler (Essen), H. Pahlings (Aachen) and C.M. Ringel (Bielefeld) in order to strengthen the interaction between the different research areas in representation theory. The Deutsche Forschungsgemeinschaft has given many research positions and fellowships for young algebraists enabling them to do research at their own universities or as visitors at well known research institutions in America, Australia, England and France. The whole project benefitted very much from an extensive exchange programme between German and American scientists sponsored by the Deutsche Forschungsgemeinschaft and by the National Science Foundation of the United States. This volume presents lectures given in a final conference and reports by members of the project. It is divided into two parts. The first part contains seven survey articles describing recent advances in different areas of representation theory. These articles do not only concentrate on the work done by the German research groups, but also inform on major developments of the subject at all. The volume omits those topics already treated in book form. In particular, it does not contain a survey on K. Erdmann's work on the structure of tame blocks of finite groups; her recent book explains in detail the fruitful interaction between the homological and combinatorial methods of the representation theory of finite dimensional algebras, and the group and character theoretical methods of group representation theory. D. Happel's book on triangulated categories and the representation theory of algebras and the book on Brauer trees written by G. Hiss and K. Lux present important recent developments. These books have already become standard references. The second part contains 17 articles representing the contents of the lectures which have be delivered at the conference on "Representation Theory of Finite Groups and Finite Dimensional Algebras" at the University of Bielefeld from 15 until 17 May Invited lectures have been given by M. Auslander, J.A. Green, M. Isaacs and J.C. Jantzen. These mathematicians have influenced the whole project very much either as visitors at German universities or as hosts of German visitors at their home institutions. Their articles are followed by 13 research papers of German algebraists having benefitted from the support of the project. The editors thank the Deutsche Forschungsgemeinschaft for its support. G. Michler, C.M. Ringel Bielefeld and Essen, January 1991

8 viii CONTRIBUTORS M. Auslander C. Bessenrodt E. Dieterich R. Dipper K. Erdmann B. Fischer J.A. Green D. Happel G. Hiss B. Huppert M. Isaacs J.C. Jantzen R. Knorr B. K iilshammer K. Lux G. Malle O. Manz Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254, USA Institut fiir Experimentelle Mathematik, Universitiit GHS Essen, 4300 Essen 12, Germany Mathematisches Institut, Universitiit Ziirich, 8001 Ziirich, Switzerland Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA Somerville College, University of Oxford, Oxford OX2 6HD, England Fakultiit fiir Mathematik, Universitiit Bielefeld, 4800 Bielefeld 1, Germany Mathematics Institute, University of Warwick, Coventry CV 47 AL, England Fakultiit fiir Mathematik, Universitiit Bielefeld, 4800 Bielefeld 1, Germany Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen, Universitiit Heidelberg, 6900 Heidelberg, Germany Fachbereich Mathematik, Universitiit Mainz, 6500 Mainz, Germany Department of Mathematics, University of Wisconsin, Madison, Wisconsin , USA Department of Mathematics, University of Oregon, Eugene, Oregon , USA Fachbereich Mathematik, Universitiit Mainz, 6500 Mainz, Germany Institut fiir Mathematik, Universitiit Augsburg, 8900 Augsburg, Germany Lehrstuhl D fiir Mathematik, Technische Universitiit, 5100 Aachen, Germany Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen, Universitiit Heidelberg, 6900 Heidelberg, Germany Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen, Universitiit Heidelberg, 6900 Heidelberg, Germany

9 ix B.H. Matzat G.O. Michler H. Pahlings W. Plesken I. Reiten C.M. Ringel K.W. Roggenkamp A. Wiedemann W. Willems Interdiszipliniires Zentrum fiir Wissenschaftliches Rechnen, Universitat Heidelberg, 6900 Heidelberg, Germany Institut fiir Experimentelle Mathematik, Universitat GHS Essen, 4300 Essen 12, Germany Lehrstuhl D fiir Mathematik, Technische Universitat, 5100 Aachen, Germany Lehrstuhl B fiir Mathematik, Technische Universitat, 5100 Aachen, Germany Institutt for Matematikk og Statistikk, Universitetet i Trondheim, 7055 Dragvoll, Norway Fakultat fiir Mathematik, Universitat Bielefeld, 4800 Bielefeld 1, Germany Mathematisches Institut B, Universitat Stuttgart, 7000 Stuttgart 80, Germany Mathematisches Institut B, Universitat Stuttgart, 7000 Stuttgart 80, Germany Institut fiir Experimentelle Mathematik, Universitat GHS Essen, 4300 Essen 12, Germany

Dan Zacharia Curriculum Vitae

Dan Zacharia Curriculum Vitae Department of Mathematics Syracuse University Syracuse, NY 13244 Telephone: (315) 443-1580 (Office) (315) 443-1475 (Fax) (315) 445-9171 (Home) Email: zacharia@syr.edu Nationality: