Investigations in Algebraic Theory of Combinatorial Objects

513 Pages·1994·19.181 MB·English

by  I. A. Faradžev, M. H. Klin, M. E. Muzichuk (auth.), I. A. Faradžev, A. A. Ivanov, M. H. Klin, A. J. Woldar (eds.)

Investigations in Algebraic Theory of Combinatorial Objects
Mathematics and lts Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, Russia
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, Russia
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, Russia
S. P. NOVIKOV, Landau Institute ofTheoretical Physics, Moscow, Russia
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, Russia
Volume 84
Investigations in
Algebraic Theory of
Combinatorial Objects
Edited by
I. A. Faradzev
A. A. Ivanov
M. H. Klin
Institute for System Studies,
Moscow, Russia
and
A. J. Woldar
Villanova University,
Villanova, Pennsylvania, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Issledovanifa po algebralchesko1 teorii kambinatornykh ob"ektov.
English.
Investigt1ons in algebraic theory of combinatorial Objects I by
I.A. Faradzev, A.A. Ivanov, M.H. Kl in. and A.J. Waldar.
p. cm. -- <Mathematics and its applications. Soviet series
84>
Inc 1 udes i ndex.
ISBN 978-90-481-4195-1 ISBN 978-94-017-1972-8 (eBook)
DOI 10.1007/978-94-017-1972-8
1. Combinatorial analysis--Congresses. I. Faradzhev, I. A.
II. Title. III. Ser1es: Mathmematics and its appl ications <Kluwer
Academ1c Publ ishers>. Soviet ser1es ; 84.
QA164.I8713 1992
511 ·. 6--dc20 92-27720
ISBN 978-90-481-4195-1
Part of this book is a revised and updated translation of
HCCJ1E,LJ,OBAHH51 no AJlfEBPAH4ECKOA TEOPHH
KOMBHHATOPHblX OBbEKTOB
© Institute for System Studies, Moscow, 1985
Printedon acid-free paper
All Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in I 994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
Preface to the English edition vii
Preface to the Russian edition ix
PART 1. CELLULAR RINGS
1.1 I.A. Faradrev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of
automorphisms of graphs
1.2 V.A. Ustimenko, Onp-local analysis ofpermutation groups 153
l.3 J a.Ju. Gol'fand, A. V. Ivanov, M.H. Klin, Amorphie cellular rings 167
1.4 M.E. Muzichuk, The subschemes ofthe Hamming scheme 187
1.5 Ja.Ju. Gol'fand, A description of subrings in V(SP, X sP, X ••• X Sp) 209
1.6 LA. Faradrev, Cellular subrings of the symmetric square of a cellular ring of rank 3 225
1.7 V.A. Ustimenko, The intersection numbers of the Hecke algebras H(PGL.(q),
BWß) 251
1.8 LA. Faradiev, A.V. Ivanov, Ranksand subdegrees of the symmetric groups acting
on partitions 265
1.9 A.A. Ivanov, Computation of lengths of orbits of a subgroup in a transitive
permutation group 275
PART 2. DISTANCE-TRANSITIVE GRAPHS
2.1 A.A. Ivanov, Distance-transitive graphs and their classification 283
2.2 A.V. Ivanov, On some local characteristics of distance-transitive graphs 379
2.3 LV. Chuvaeva, A.A. Ivanov, Action ofthe groupM on Hadamard matrices 395
12
2.4 F.L. Tchuda, Construction of an automorphic graph on 280 vertices using finite
geometries 409
PART 3. AMALGAMSAND DIAGRAM GEOMETRIES
3.1 A.A. Ivanov, S.V. Shpectorov, Applications of group amalgams to algebraic graph
theory 417
3.2 S.V. Shpectorov, A geometric characterization of the group M 443
22
3.3 M.E. lofmova, A.A. Ivanov, Bi-primitive cubic graphs 459
3.4 V.A. Ustimenko, On some properties of geometries of Chevalley groups and their
generalizations 473
Subject index 507
PREFACE TO TUE ENGLISH EDITION
This volume arose through the initiative of Kluwer Academic Publishers in an attempt to intro
duce some areas of research in algebraic combinatorics which originally appeared in Russian to a
wider mathematical community. The authors of the papers in this volume belong to two scientific
groups. The first consists of the people associated with the Iabaratory of Discrete Mathematics at
the Institute of System Studies in Moscow. The other belongs to the Department of Algebra at the
Kiev State University. Besides translations of research papers from Russian to English, the
volume contains four survey papers written expressly for this edition. The surveys are located in
the opening sections of each of the parts of the volume: two surveys belong to the first part, one
to the second and one to the third.
The core of the volume is formed by the c:ollection of papers "Investigations in Algebraic
Theory of Combinatorial Objects" (M.H. Klin, I.A. Faradrev eds.), Moscow, Institute for System
Studies, 1985, referred below as IATC0-85. For the papers translated from IATC0-85 we
indicate the corresponding pages at the end of the papers. The present volume contains transla
tions of all papers from IATC0-85 excepting the first and the last ones. The content of the first
paper is covered in the survey "Cellular Rings and Groups of Automorphisms of Graphs" by LA.
Faradtev, M.H. Klin and M.E. Muzichuk. For this reason tlle references in the translated papers
to the first paper of IATC0-85 were changed to the references to the above mentioned survey. On
the other hand, the volume contains a translatiort of a paper by A.A. lvanov on the computation of
ranks and subdegrees in permutation groups, which appeared originally in a different collection.
The volume consists of three parts. The papers in the first part are devoted to investigations and
applications of cellular rings (adjacency algebras of coherent configurations). The first survey in
this part is about the subject in general. The second survey is on the technique of p-local analysis
in permutation groups. The second part of the volume contains papers on distance-regular and
distance-transitive graphs. In the third part some results in a relatively new direction, amalgams
and geometries, are presented. As was predicted in the preface to the Russian edition, the method
of amalgams has come to play an increasingly significant roJe in algebraic combinatorics.
Some changes were made to the papers from the Russian collection during translation. The
paper "On some properties of the geometries of the Chevalley groups and their generalizations"
by V.A. Ustimenko was revised considerably. The paper "Amorphie cellular rings" by Ja.Ju.
Gol'fand, A.V. Ivanov and M.H. Klin in the Russian editionwas divided into two 'parts', with
different sets of authors. Finally in some papers the order of the authors was changed to be
alphabetic in the Western version.
The Publisher wishes to draw the readers' attention to the special issue of the Kluwer joumal
Acta Applicandae Mathematicae Vol. 29/1-2 entitled "Interactions between Algebra and Com
binatorics", edited by LA. Faradrev, A.A. Ivanov, and M. H Klin. This issue can be seen as a
sequel to the present volume. It deals with cellular rings, distance-regular graphs and group
vii
viii PREFACE TO THE ENGLISH EDITlON
amalgams; the papers provide examples of new applications of permutation group theory and
association schemes in algebraic combinatorics.
We are very grateful to Professor M. Hazewinkel for his interest in our research and for
introducing the idea for the present volume, to Dr. D.J. Lamer for his help and patience during
the delayed preparation of the volume, and to Adrianka de Wit and Anneke Pot for their kind and
perfect technical assistance.
We want to thank J. Remmeterand F. Lazebnik for their excellent and conscientious transla
tiontagether with the fourth editor of the massive paper "Cellular Rings and Groups of Automor
phisms of Graphs".
I.A. Faradzev
A.A. Ivanov
M.H. Klin
A. Waldar
PREFACE TO TUE RUSSIAN EDITION
In its development modern combinatorics synthesized methods from many diverse branches of
mathematics, especially from algebra, geometry and number theory. The main roJe in this
synthesis was played by an extant body of algebraic ideas, starting primarily with techniques
from linear algebra and group theory.
Over the last twenty years a new tendency developed toward the interplay of combinatorial and
algebraic (particularly group theoretic) methods .. This tendency was caused by a sharp increase in
the importance of discrete mathematical applications. It was discovered that certain problems
related to experimental design, chemical structure analysis, design of logical schemes and of
various devices, etc., shared a common mathematical formulation. The content of these problems
is the identification of certain combinatorial objects (primarily graphs and networks) and the
characterization of their automorphism groups.
In the mid 70s, a variety of results from 11nite group theory on automorphism groups of
combinatorial objects, tagether with combinatorics and computing theory, formed the basis for
the direction and content of a new mathematical subject. Subsequently, the algebraic theory of
combinatorial objects became an independent branch of mathematics. The main goal of this
theory is to study the relationship between a combinatorial object's local features, defined in
terms of incidence of its component parts (e.g. vertices and edges of a graph; points, bloclcs and
flags of a block design), and the global properties of the object's automorphism group. From the
group theoretic viewpoint, interest in such a relationship is justified by the compactness and
convenience by which one is able to defme certain classes of groups as automorphism groups of
graphs, two-graphs and other appropriate combinatorial objects. The use of such definitions
appears to have been extremely efficient in studying the sporndie simple groups, for example.
Exploiting the relationship between local and global properties enables one to establish
necessary (and sometimes sufficient) conditions for the automorphism group of an object from a
certain class to have such extremal properties as transitivity, primitivity, distance-transitivity, etc.
From the viewpoint of complex systems theory, the primary objective of the algebraic theory of
combinatorial objects is to approximate the global (i.e. algebraic) properlies of a system by its
local (i.e. combinatorial) features, that is, features which involve incidence between its com
ponent parts.
The present collection of papers deals with an area of the subject that became uni11ed only a
few years ago. This area is founded on the following three methodological bases, which arose
independently of one another: the method of invariant relations, the theory of cellular rings, and
constructive enumeration of combinatorial objects. The oldest among these is the method of
invariant relations in the theory of permutation groups. This method was proposed by M. Krasner
and I. Schur in the 30s and was advanced fundamentally in papers by H. Wielandt, R.
ix
X PREFACE TO THE RUSSIAN EDITION
Köchendorffer, L.A. Kalu:lnin and their students in the 50s and 60s. Nowadays the most deeply
developed is the theory of binary invariant relations and their combinatorial approximations.
These combinatorial approximations arose repeatedly during this century under various names
(Hecke algebras, centralizer rings, association schemes, coherent configurations, cellular rings,
etc.-see the first paper of the collection for details) andin various branches of mathematics, both
pure and applied. One of these approximations, the theory of cellular rings (cellular algebras),
was developed at the end of the 60s by B. Yu. Weisfeiler and A.A. Leman in the course of the
first serious attempt to study the complexity of the graph isomorphism problem, one of the central
problems in the modern theory of combinatorial algorithms. At roughly the same time G.M.
Adelson-Velskir, V.L. Arlazarov, I.A. Faradtev and their colleagues had developed a rather
efficient tool for the constructive enumeration of combinatorial objects based on the branch and
bound method. By means of this tool a number of "sports-like" results were obtained. Some of
these results are still unsurpassed.
Toward the end of the 70s it became clear that an extensive knowledge of finite group theoretic
techniques was desireable in order to obtain important new results by analyzing the enormaus
amount of data on graphs and incidence systems whose automorphism groups had interesting
properties. It was also desirable to have some powerful software available so that one could carry
out computations in permutation groups and cellular rings in order to collect this data.
The necessity to unite the efforts of specialists from different scientific schools led to the
organization of a seminar on the algebraic theory of combinatorial objects. This seminar started in
1980. Its kerne! consisted of mathematicians from the Labaratory of Discrete Mathematics at the
Institute for System Studies and from the algebraic school of L.A. Kalu:lnin at the Department of
Algebra and Logic at the Kiev State University named after T.G. Shevchenko. In addition the
seminarwas attended by mathematicians from other institutions: the Department of Algebra and
Geometry at the Kaluga Pedagogical Institute named after K.E. Tsiolkovskir. the Department of
Mathematics and Mechanics and the Department of Chemistry at the Moscow State University
named after M.V. Lomonosov, the Moscow Physical-Technical Institute, etc.
The results presented in this collection were obtained through the interaction of the participants
of the seminar. The papers are divided into two parts. Those in the first part are devoted to
various aspects of cellular rings: axiomatics, the description of cellular rings possessing certain
extremal properties, enumeration of cellular rings and computation of their structure constants.
The first part opens with a survey paper by M.H. Klin which contains almost no new results but
presents a history and methodology of the subject, as weil as an introduction to terminology
requisite for an understanding of subsequent papers in the first part. The second part is devoted to
the study of automorphism groups of certain combinatorial objects such as diagram geometries,
distance-regular graphs, edge- but not vertex-transitive graphs, Hadamard matrices and structure
formulas of chemical compounds. This part opens with a paper by S.V. Shpectorov, in which a
new method for the characterization of certain combinatorial objects (diagram geometries and
graphs) is proposed which is based on a consideration of rank 3 amalgams. The amalgam method