Theory of Association Schemes
The present text is an introduction to the theory of association schemes. We start with the de?nition of an association scheme (or a scheme as we shall say brie?y), and in order to do so we ?x a set and call it X. We write 1 to denote the set of all pairs
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Paul-Hermann Zieschang
Theory of Association Schemes
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Paul-Hermann Zieschang Department of Mathematics University of Texas at Brownsville Brownsville, TX 78520 USA E-mail: [email protected]
Library of Congress Control Number: 2005930450 Mathematics Subject Classification (2000): 05E30, 20N99 ISSN 1439-7382 ISBN-10 3-540-26136-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26136-0 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 springeronline.com c 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 protective laws and regulations and therefore free for general use. Typesetting: by the author and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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Preface
The present text is an introduction to the theory of association schemes. We start with the definition of an association scheme (or a scheme as we shall say briefly), and in order to do so we fix a set and call it X. We write 1X to denote the set of all pairs (x, x) with x ∈ X. For each subset r of the cartesian product X × X, we define r∗ to be the set of all pairs (y, z) with (z, y) ∈ r. For x an element of X and r a subset of X × X, we shall denote by xr the set of all elements y in X with (x, y) ∈ r. Let us fix a partition S of X × X with ∅ ∈ / S and 1X ∈ S, and let us assume that s∗ ∈ S for each element s in S. The set S is called a scheme on X if, for any three elements p, q, and r in S, there exists a cardinal number apqr such that |yp ∩ zq ∗ | = apqr for any two elements y in X and z in yr. The notion of a scheme generalizes naturally the notion of a group, and we shall base all our considerations on this observation. Let us, therefore, briefly look at the relationship between groups and schemes. Let S be a scheme, and let P and Q be nonempty subsets of S. We define P Q to be the set of all elements s in S for which there exist elements p in P and q in Q satisfying 1 ≤ apqs . If P possesses an element p with {p} = P and Q an element q satisfying {q} = Q, we write pq instead of P Q. The set P Q will be called the complex product of P and Q. The associated operation on the set of all nonempty subsets of S will be
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