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Báo cáo toán học: "Separability Number and Schurity Number of Coherent Configurations"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Separability Number and Schurity Number of Coherent Configurations. | Separability Number and Schurity Number of Coherent Configurations Sergei Evdokimov St. Petersburg Institute for Informatics and Automation evdokim@pdmi.ras.ru Ilia Ponomarenko Steklov Institute of Mathematics at St. Petersburg inp@pdmi.ras.ru t Submitted January 26 2000 Accepted May 17 2000 Abstract To each coherent configuration scheme C and positive integer m we associate a natural scheme c m on the m-fold Cartesian product of the point set of C having the same automorphism group as C. Using this construction we define and study two positive integers the separability number s C and the Schurity number t C of C. It turns out that s C m iff C is uniquely determined up to isomorphism by the intersection numbers of the scheme C- . Similarly t C m iff the diagonal subscheme of C is an orbital one. In particular if C is the scheme of a distance-regular graph r then s C 1 iff r is uniquely determined by its parameters whereas t C 1 iff r is distance-transitive. We show that if C is a Johnson Hamming or Grassmann scheme then s C 2 and t C 1. Moreover we find the exact values of s C and t C for the scheme C associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular s C t C 2 if C is the scheme of a Doob graph. In addition we prove that s C 2 and t C 2 for any imprimitive 3 2-homogeneous scheme. Finally we show that s C 4 whenever C is a cyclotomic scheme on a prime number of points. 1 Introduction The purpose of this paper is to continue the investigations of distance-regular graphs 4 and more generally association schemes 3 from the point of view of their isomorphisms Partially supported by RFFI grant 96-15-96060 tPartially supported by RFFI grants 96-15-96060 99-01-00098 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R31 2 and symmetries started by the authors in 9 11 12 . We have tried to make this paper self-contained but nevertheless some knowledge of basic algebraic combinatorics in the spirit of the .