Representations
It will turn out to be useful to refine the preceding considerations of permutation representations. We consider linear representations on corresponding vector spaces and decompose them into their irreducible constituents. Having done this we can use all
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Algorithms and Combinatorics 19
Editorial Board R.L. Graham, Murray Hill B. Korte, Bonn L. Lovasz, Budapest A.Wigderson, Jerusalem G.M. Ziegler, Berlin
Springer-Verlag Berlin Heidelberg GmbH
Adalbert Kerber
Applied Finite Group Actions
2nd, Revised and Expanded Edition
i
Springer
Adalbert Kerber University of Bayreuth Department of Mathematics 95440 Bayreuth Germany kerber@uni-bayreuth. de
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kerber, Adalbert: Applied finite group actions I Adalbert Kerber. - 2., rev. and expanded ed. (Algorithms and combinatorics; 19) l. Aufl. im BI-Wiss.-Verl. u.d.T.: Kerber, Adalbert: Algebraic combinatorics via finite group actions ISBN 978-3-662-11167-3 (eBook) ISBN 978-3-642-08522-2 DOI 10.1007/978-3-662-11167-3
The first edition of this book was published by BI Wissenschaftsverlag in 1991
Mathematics Subject Classification (1991): osExx, 20-01, 20-02, 20B30, 20B35, 20C15, 20C30 ISSN 0937-ssn ISBN 978-3-642-08522-2 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 microfilms 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-VerlagBerlinHeidelbergGmbH. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999 Softcover reprint of the hardcover 2nd edition 1999 Computer to film: Mercedesdruck, Berlin Typesetting: Typeset in LaTEX by the author. 41/3143- 54 3 2 1 o- Printed on acid-free paper SPIN 10654673
To Dieta
Preface to the Second Edition
Also the present second edition of this book is an introduction to the theory of classification, enumeration, construction and generation of finite unlabeled structures in mathematics and sciences. Since the publication of the first edition in 1991 the constructive theory of unlabeled finite structures has made remarkable progress. For example, the first 7designs with moderate parameters were constructed, in Bayreuth, by the end of 1994 ([9]). The crucial steps were - the prescription of a suitable group of automorphisms, i. e. a stabilizer, and the corresponding use of Kramer-Mesner matrices, together with - an implementation of an improved version of the LLL-algorithm that allowed to find 0-1-solutions of a system of linear equations with the Kramer-Mesner matrix as its matrix of coefficients. The Kramer-Mesner matrices can be considered as submatrices of matrices of the form A" (see the chapter on group actions on posets, semigroups and lattices). They are associated with the action of the prescribed group G which is a permutation group on a set X of points i
