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Encyclopaedia of Mathematical Sciences Volume 135 Invariant Encyclopaedia of Mathematical Sciences Volume 135 Invariant Theory and Algebraic Transformation Groups VI Subseries Editors: R.V. Gamkrelidze V.L. Popov Martin Lorenz Multiplicative Invariant Theory 123 Author Martin Lorenz Department of Mathematics Temple Universit y Philadelphia, PA 19122, USA e-mail: [email protected] Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze Mathematics Subject Classification (2000): Primary: 13A50 Secondary: 13H10, 13D45, 20C10, 12F20 ISSN 0938-0396 ISBN 3-540-24323-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media GmbH springeronline.com ©Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover Design: E. Kirchner, Heidelberg, Germany Printed on acid-free paper 46/3142 YL 5 4 3 2 1 0 To my mother, Martha Lorenz, and to the memory of my father, Adolf Lorenz (1925 – 2001) Preface Multiplicative invariant theory, as a research area in its own right, is of relatively recent vintage: the systematic investigation of multiplicative invariants was initiated by Daniel Farkas in the 1980s. Since then the subject has been pursued by a small but growing number of researchers, and at this point it has reached a stage in its devel- opment where a coherent account of the basic results achieved thus far is desirable. Such is the goal of this book. The topic of multiplicative invariant theory is intimately tied to integral repre- sentations of finite groups. Therefore, the field has a predominantly discrete, alge- braic flavor. Geometry, specifically the theory of algebraic groups, enters the picture through Weyl groups and their root lattices as well as via character lattices of alge- braic tori. I have tried to keep this book reasonably self-contained. The core results on mul- tiplicative invariants are presented with complete proofs often improving on those found in the literature. The prerequisites from representation theory and the theory of root systems are assembled early in the text, for the most part with references to Curtis and Reiner [44], [45] and Bourbaki [24]. For multiplicative invariant algebras, Chapters 3–8 give an essentially complete account of the state of the subject to date. On the other hand, more is known about multiplicative invariant fields than what found its way into this book. The reader may wish to consult the monographs by Saltman [186] and Voskresenski˘ı [220] for additional information. A novel feature of the present text is the full and streamlined derivation of the known rationality properties of the field of matrix invariants in Chapter 9. This material could heretofore only be found in the original sources which are widely spread in the literature. A more detailed overview of the contents of this book is offered in the Intro- duction below. In addition, each of the subsequent chapters has its own introductory section; those of Chapters 4–9 are quite extensive, giving complete statements of the main results proved and delineating the pertinent algebraic background. The book concludes with a chapter on research problems. I hope that it will stimulate further interest in the field of multiplicative invariant theory. VIII Preface Acknowledgements Dan Farkas’ early work on multiplicative invariants ([59], [60], [61]) provided the initial impetus for my own research on the subject. I also owe an enormous debt to Don Passman at whose invitation I spent a postdoctoral year at UW-Madison in 1978/9 and who has been a good friend ever since. In many ways, I have tried to em- ulate his book “Infinite Group Rings” [147] in these notes. More recently, I have ben- efitted from the insights of Nicole Lemire and Zinovy Reichstein, both through their own work on multiplicative invariant theory and through collaborations on related topics. For comments on earlier drafts and help with various aspects of this book, I wish to thank Jacques Alev, Esther Beneish, Frank DeMeyer, Steve Donkin, Victor Guba, Jens Carsten Jantzen, Gregor Kemper, Gabriele Nebe, Wilhelm Plesken, and David Saltman. I am grateful to Vladimir Popov for inviting me to write this account of my fa- vorite research area and to the anonymous referees for their constructive suggestions. While writing this book, I was supported in part by the National Science Foun- dation under grant DMS-9988756 and by the Leverhulme Foundation (Research In- terchange Grant F/00158/X). My greatest debt is to my family: to my daughter Esther for her unwavering support over many years and long distances and to my wife Maria and our chil- dren, Gabriel, Dalia and Aidan, whose loving and energetic presence is keeping me grounded. Temple University, Philadelphia Martin Lorenz December 2004 Contents Introduction ....................................................... 1 Notations and Conventions .......................................... 9 1 Groups Acting on Lattices ...................................... 13 1.1 Introduction ............................................... 13 1.2 G-Lattices ................................................ 13 1.3 Examples ................................................. 14 1.4 Standard Lattice Constructions . ............................. 16 1.5 Indecomposable Lattices .................................... 18 1.6 Conditioning the Lattice . .................................... 20 1.7 Reflections and Generalized Reflections ........................ 21 1.8 Lattices Associated with Root Systems ........................ 24 1.9 The Root System Associated with a Faithful G-Lattice............ 27 1.10 Finite Subgroups of GLn( ) ................................. 28 2 Permutation Lattices and Flasque Equivalence .................... 33 2.1 Introduction ............................................... 33 2.2 PermutationLattices........................................ 33 2.3 Stable Permutation Equivalence . ............................. 34 2.4 PermutationProjectiveLattices............................... 35 2.5 H i-trivial, Flasque and Coflasque Lattices ...................... 36 2.6 Flasque and Coflasque Resolutions ............................ 37 2.7 Flasque Equivalence ........................................ 38 2.8 Quasi-permutation Lattices and Monomial Lattices . ........... 39 2.9 An Invariant for Flasque Equivalence . ......................... 40 2.10 Overview of Lattice Types . ............................. 42 2.11 Restriction to the Sylow Normalizer ........................... 43 2.12 Some Sn-Lattices.......................................... 44 X Contents 3 Multiplicative Actions .......................................... 51 3.1 Introduction ............................................... 51 3.2 The Group Algebra of a G-Lattice . ...................... 51 3.3 Reduction to Finite Groups, -structure, and Finite Generation . 52 3.4 Units and Semigroup Algebras ............................... 53 3.5 Examples ................................................. 55 3.6 Multiplicative Invariants of Weight Lattices . .................. 61 3.7 Passage to an Effective Lattice . ............................. 63 3.8 Twisted Multiplicative Actions . ............................. 64 3.9 Hopf Structure ............................................. 66 3.10TorusInvariants............................................ 67 4 Class Group ................................................... 69 4.1 Introduction ............................................... 69 4.2 Some Examples . ........................................... 70 4.3 Krull Domains and Class Groups ............................. 70 4.4 Samuel’s Exact Sequence.................................... 72 4.5 Generalized Reflections on Rings ............................. 73 4.6 Proof of Theorem 4.1.1 ...................................... 75 5 Picard Group .................................................. 77 5.1 Introduction ............................................... 77 5.2 Invertible Modules . ........................................ 78 5.3 TheSkewGroupRing ...................................... 79 5.4 TheTraceMap............................................. 80 5.5 The Kernel of the Map Pic(RG) → Pic(R) .................... 81 5.6 The Case of Multiplicative Actions . ...................... 83 6 Multiplicative Invariants of Reflection Groups ..................... 85 6.1 Introduction ............................................... 85 6.2 Proof of Theorem 6.1.1 ...................................... 86 6.3 Computing the Ring of Invariants ............................. 87 6.4 SAGBIBases.............................................. 91 7 Regularity ....................................................
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