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插图：The concept of the association scheme together with partially balanced incomplete blockdesigns was defined in its own right by R. C. Bose and T. Shimamoto in 1952. It wasintroduced to describe the balance relations among the treatments of partially balanced in-complete block designs. Association schemes have close connections with coding theory,graph theory, and finite group theory, and in particular, provide a framework for studyingcodes and designs. By the 1980s, association scheme theory was an important branch ofalgebraic combinatorics, and the research work on association scheme theory had growntremendously.The study of association schemes in China was started by Professors L. C. Chang andPao-Lu Hsu in the late 1950s. Later, my students and I began to construct associationschemes and block designs using various subspaces of vector spaces under the action ofclassical groups. These results were collected in the monograph Studies in Finite Geome-tries and the Construction of Partially Incomplete Block Designs by Z. Wan, Z. Dai, X.Feng, and B. Yang published by Science Press （Beijing, 1966）. In the mid-1960s, I con-strutted a family of association schemes on Hermitian matrices and computed the param-eters of the lower dimensional ones [20] and started a new direction of construction ofassociation schemes on matrices. The association scheme theory developed later indicatesthat the association schemes of maximal totally isotropic subspaces and of Hermitian ma-trices are known as primitive P- and Q-polynomial association schemes.In the late 1970s, Professor Yangxian Wang continued the study of association schemesof matrices. He derived formulas for the parameters of association schemes of Hermitianmatrices and constructed association schemes using rectangular matrices and alternate ma-trices. Later, Professors Yuanji Hut, Xueli Zhu and I studied the association schemes ofsymmetric matrices in odd characteristic. In the 1990s, Professor Yangxian Wang and hisstudents Jianmin Ma and Changli Ma at that time studied the association schemes of sym-metric matrices and quadratic forms in even characteristic. Besides the parameters of theseassociation schemes, they discussed the subschemes, quotient schemes, and duality andautomorphisms. S.o the study of association schemes of matrices has reached a more com-plete stage. In this monograph, Professors Yangxian Wang, Yuanji Hut, and Dr. ChangliMa collect the results on association schemes of matrices in a systematic way. The aimof this monograph is to study the association schemes of matrices, including construction,parameter calculation, primitivity, duality, automorphisms and polynomial properties, etc.I hope this monograph will provide readers with some methods and tools to study associ-ation schemes and bring new results.

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自然科学 数学 代数数论组合理论

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