Algorithms in Invariant Theory (2nd ed.) [Sturmfels 2008-06-06].pdf

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Texts and Monographs in
Symbolic Computation
A Series of the
Research Institute for Symbolic Computation,
Johannes Kepler University, Linz, Austria
Edited by P. Paule
Bernd Sturmfels
Algorithms in Invariant Theory
Second edition
SpringerWienNewYork
Dr. Bernd Sturmfels
Department of Mathematics
University of California, Berkeley, California, U.S.A.
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SPIN 12185696
With 5 Figures
Library of Congress Control Number 2007941496
ISSN 0943-853X
ISBN 978-3-211-77416-8 SpringerWienNewYork
ISBN 3-211-82445-6 1st edn. SpringerWienNewYork
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 Gröbner bases;
(ii)
Combinatorial algorithms
in invariant theory, such as the straightening al-
gorithm, which relate to representation theory of the general linear group;
(iii)
Applications
to projective geometry.
Part of this material was covered in a graduate course which I taught at RISC-
Linz 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. Laksh-
man, 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 beau-
tiful subject of invariant theory, and for collaborating with me on the topics in
Chapters 2 and 3. I am grateful to the following institutions for their support: the
Austrian Science Foundation (FWF), the U.S. Army Research Office (through
MSI Cornell), the National Science Foundation, the Alfred P. Sloan Foundation,
and the Mittag-Leffler Institute (Stockholm).
Ithaca, June 1993
Bernd Sturmfels
(i)

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