On balanced sets and related structures (Algebraic Combinatorics)
Hiroshi Suzuki · Institutional Repositories DataBase (IRDB) · 1999
MotivationI have been studying distance-regular graphs since 1990.Distance-regular graphs are also known as $P$ -polynomial association schemes as there is a natural correspondence be- tween them.Distance-regular graphs are defined to satisfy ideal regularity conditions, and there are lots of excellent researches using the combinatorial regularity.(See for example [10,11,17,21].)Moreover, these developments in the study of combinatorial aspect of distance-regular graphs have been successfully applied to develop the theory of association schemes which are not necessarily $P$ -polynomial ([12, 24, 25, 26]).On the other hand, association schemes have both combinatorial and algebraic structures, it is natural to use algebraic properties of the association scheme associated to each distance-regular graph.However, besides the use of the integrality condition of multiplicities of eigenvalues of the adjacency matrix, the only analysis successfully applied using the algebraic structures is made under extra algebraic condition such as $Q$ -polynomial property.I strongly feel the need of the development of the study on the algebraic properties of association schemes and their representation theory.As the combinatorial theory is developing with the combinatorial analysis of $P$ -polynomial association scheme, which has the ideal combinatorial structure among association schemes, I feel that it should be very important to study al- gebraic properties and their representation theory of association schemes which have ideal algebraic condition.$Q$ -polynomial association schemes and balanced conditions which de- fine $Q$ -polynomial property of distance-regular graphs seem to be the structure we should study first ([18, 19]).In $1970' \mathrm{s},$ $Q$ -polynomial association schemes were defined as schemes 'dual' in a sense to $P$ -polynomial association schemes by P. Delsarte and they were studied in connection with the design theory and the tight condition ([4]).In $1980' \mathrm{s}$ , P. Terwilliger introduced balanced conditions in order to describe the $Q$ -polynomial properties of distance-regular graphs ([22, 23]).He defined balanced condition and strongly balanced condition.He showed that a distance-regular graph satisfies the balanced condition if and only if the association scheme is $Q$ -polynomial.In $1990' \mathrm{s}$ , P. Terwilliger with the aid of his student G. Dickie classified all distance-regular graphs with strongly balanced condition ([5, 6]).These are all excellent results, but the general theory of $Q$ -polynomial association schemes and that of balanced conditions have not been much studied.For example, we do not have many examples of $Q$ -polynomial association schemes, which are not P- polynomial.(Most of the classical $P$ -polynomial association schemes are Q-polynomial