Structures and Representations of Association Schemes

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a. Absolute Bound Conjecture of Distance-Regular Graphs and Related Topics: Good progress was made in the classification of distance-regular graphs of order (s, 2). Recent results were included in a lecture note.b. Q-Polynomial Schemes and Balanced Sets: New conditions on Krein parameters were given and several long standing problems of Q-polynomial association schemes were solved u applications of these conditions.c. Decompositions of the Character Products of Finite Croups: The case was studied that the number of distinct irreducible constituents of χィイD12ィエD1, the square of an irreducible character, χ is small. As an application, complete classification of Q-polynomial group schemes was made in cooperation with M. Kiyota.d. Research on Commutative Association Schemes:(1) Spin Models: Recently, Spin models have been studied by many as link invariants. They are defined by a pair of type II matrices, and each type II matrix is associated with a so-called Nomura algebra, which is the Bo … More se-Mesner algebra of a commutative association scheme. As a joint work with H. Tsuchiyama, the complete classification of spin models of size 6 and 7, using the connection between type II matrices and Bose-Mesner algebras of commutative association schemes. Very recently, as a joint work With Rie Hosoya, it was shown that if the Nomura algebra is imprimitive, the type II matrix can be decomposed as a generalized tensor product of type II matrices of smaller size.(2) Balanced Conditions: Desirable generalization of the definition of balanced conditions of non-symmetric commutative association schemes was obtained.(3) Weakly Distance-Regular Digraphs: New and more natural definition of distance regular digraphs, which were called here weakly distance-regular digraphs, was given. As a joint work with Kaishun Wang, the classification of the case when the degree is 2 was completed very recently.e. International Symposium on Algebraic Combinatorics was held at International Christian University in July l997. As an organizer, the symposium home page was prepared and it became 'Algebraic Combinatorics Home Page' (http://alcom.icu.ac.jp/) using the computer installed by the grant. Less