Bannai Dimension Problem Over The Johnson Scheme

Johnson scheme is one of the most important association scheme in Algebraic combinatorics. It is a combination of a typical representative of the association scheme. Over the association scheme can define a function space. The function space over Johnson scheme has a deep and broad applications on the combination of design, especially for the combination of design on the sphere and Euclidean combination of design. However,research of this function space is still in its initial stages due to the intrinsic complexity of the association scheme. In order to understand the structure of the function space on Johnson scheme, one first needs to obtain the dimension of these function spaces. Professor Eiichi Bannai particular emphasizes the importance and significance of the function space dimension over the association scheme at Shanghai Jiao Tong University Spring2012"Algebraic Combinatorics Seminar". He makes a number of conjectures about these dimensions.Through the large number of examples and calculations, Bannai dimension conjecture of the Johnson Scheme is proved in this paper.The main contents of this paper include:The first part describes the relevant background knowledge. A brief introduction to the history and development of algebraic combinatorics and a special presentation of the status of association scheme and the current problems are included.The second part gives the definition and basic theorems of association schemes and Johnson schemes.The third part studies the function spaces over the Johnson scheme and Bannai dimension conjecture. Through a large number of examples and calculations, the paper gives the correct form of Bannai dimension conjecture.Then it gives a complete proof on some of these situation.The fourth part gives a summary and the full text of the study need to be further expanded.