Bracket algebra and projective geometry
According to the general philosophy outlined in Sect. 1.3, analytic geometry deals with those properties of vectors and matrices which are invariant with respect to some group of linear transformations. Applying this program to projective geometry, one is
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Algorithms in Invariant Theory Bernd Sturmfels
Springer-Verlag Wien GmbH
Texts and Monographs in
Symbolic Computation
A Series of the Research Institute for Symbolic Computation, Johannes-Kepler-University, Linz, Austria Edited by B. Buchberger and G. E. Collins
Bernd Sturmfels
Algorithms in Invariant Theory
Springer-Verlag Wien GmbH
Dr. Bernd Sturmfels Department of Mathematics Cornell University, Ithaca, New York, U.S.A.
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 photo-copying machines or similar means, and storage in data banks. © 1993 Springer-Verlag Wien
Data conversion by H.-D. Ecker, Biiro fiir Textverarbeitung, Bonn Printed by Novographic, lng. Wolfgang Schmid, A-1230 Wien Printed on acid-free paper
With 5 Figures
ISBN 978-3-211-82445-0
ISBN 978-3-7091-4368-1 (eBook) DOl 10.1007/978-3-7091-4368-1
Preface The aim of this monograph is to provide an introduction to some fundamental problems, results and algorithms of invariant theory. The focus will be on the three following aspects: Algebraic algorithms in invariant theory, in particular algorithms arising from the theory of Grabner bases; (ii) Combinatorial algorithms in invariant theory, such as the straightening algorithm, which relate to representation theory of the general linear group; (iii) Applications to projective geometry.
(i)
Part of this material was covered in a graduate course which I taught at RISCLinz in the spring of 1989 and at Cornell University in the fall of 1989. The specific selection of topics has been determined by my personal taste and my belief that many interesting connections between invariant theory and symbolic computation are yet to be explored. In order to get started with her/his own explorations, the reader will find exercises at the end of each section. The exercises vary in difficulty. Some of them are easy and straightforward, while others are more difficult, and might in fact lead to research projects. Exercises which I consider "more difficult" are marked with a star. This book is intended for a diverse audience: graduate students who wish to learn the subject from scratch, researchers in the various fields of application who want to concentrate on certain aspects of the theory, specialists who need a reference on the algorithmic side of their field, and all others between these extremes. The overwhelming majority of the results in this book are well known, with many theorems dating back to the 19th century. Some of the algorithms, however, are new and not published elsewhere. I am grateful to B. Buchberger, D. Eisenbud, L. Grove, D. Kapur, Y. Lakshman, A. Logar, B. Mourrain, V. Reiner, S. Sundaram, R. Stanley, A. Zelevinsky, G. Ziegler and numerous others who supplied comments on various versions of the manuscript. Special thanks go to N. White for introducing me to the beautiful subject of invariant theory, and
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