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Theory of Association Schemes

  • Book
  • © 2005

Overview

Authors:
  1. Paul-Hermann Zieschang

    1. Department of Mathematics, University of Texas at Brownsville, Brownsville, USA

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  • Conceptual approach to association schemes is emphasized
  • First rigorous treatment of the structure theory of schemes. Schemes are considered not necessarily to be commutative or finite. As a byproduct, the theory covers (disguised as ‘Coxeter schemes’) the theory of buildings in the sense of Jacques Tits
  • Recent results such as the generalization of Sylow’s group-theoretical theorems to scheme theory and the characterization of Glauberman’s Z*-involutions in terms of scheme theory appear for the first time in book form

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (12 chapters)

  1. Front Matter

    Pages I-XV
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  2. Basic Facts

    Pages 1-16

  3. Closed Subsets

    Pages 17-38

  4. Generating Subsets

    Pages 39-62

  5. Quotient Schemes

    Pages 63-81

  6. Morphisms

    Pages 83-102

  7. Faithful Maps

    Pages 103-131

  8. Products

    Pages 133-152

  9. From Thin Schemes to Modules

    Pages 153-182

  10. Scheme Rings

    Pages 183-208

  11. Dihedral Closed Subsets

    Pages 209-236

  12. Coxeter Sets

    Pages 237-248

  13. Spherical Coxeter Sets

    Pages 249-276

  14. Back Matter

    Pages 277-283
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Keywords

About this book

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 (x,x) with x? X. For each subset X ? r of the cartesian product X×X, we de?ne 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 ?x a partition S of X×X with?? / S and 1 ? S, and let us assume X ? 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 a such pqr ? that|yp?zq| = a for any two elements y in X and z in yr. pqr 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, brie?y look at the relationship between groups and schemes.

Reviews

From the reviews:

"Theory of association schemes is a self-contained textbook. … The theory of association schemes can be applied to Hecke algebras of transitive permutation groups, and the algebras are usually noncommutative. So this treatment is also good for group theorists. … The book under review also contains many recent developments in the theory." (Akihide Hanaki, Mathematical Reviews, 2006 h)

Authors and Affiliations

  • Department of Mathematics, University of Texas at Brownsville, Brownsville, USA

    Paul-Hermann Zieschang

About the author

Paul-Hermann Zieschang received a Doctor of Natural Sciences and the Habilitation in Mathematics from the Christian-Albrechts-Universität zu Kiel. He is also Extraordinary Professor of the Christian-Albrechts-Universität zu Kiel. Presently, he holds the position of an Associate Professor at the University of Texas at Brownsville. He held visiting positions at Kansas State University and at Kyushu University in Fukuoka.

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