The structure of imprimitive non-symmetric 3-class association schemes
The authors begin to characterize the imprimitivity of non-symmetric 3-schemes (3-class association schemes) in terms of their parameters. The symmetrization of a non-symmetric 3-scheme is a symmetric 2-scheme. Let \({\mathbf R}= \{R_0,\dots, R_n\}\) be a family of binary relations on \(X\), and \((X, {\mathbf R})\) be a commutative \(n\)-scheme. Then \[ p^k_{ij}= |\{z\in X\mid (x, z)\in R_i\text{ and } (z, y)\in R_j\}| \] are independent of the choice of \((x, y)\in R_k\), and \(p^k_{ij}= p^k_{ji}\). Further, for \(R_i\in {\mathbf R}\) the inverse \(R^{- 1}_i= \{(y, x)\mid (x, y)\in R_i\}\) of the relation \(R_i\) belongs to \({\mathbf R}\) (the index of relation \(R^{- 1}_i\) is denoted by \(i^R\)). Set \(v_i= p^0_{ii_R}\) and \(u= v_1/v_2\). Theorem. For an imprimitive non-symmetric 3-scheme \((X, {\mathbf R})\), there is only one of the following possibilities: (1) \(p^1_{13}= p^1_{23}\); (2) \(up^1_{33}= p^1_{23}\); and (3) \(p^1_{11}= p^2_{11}\). There are obvious ways to construct all the schemes of types (1), (2) (Theorems 5.5 and 7.1), connected with skew-Hadamard matrices. Only partial results are found in case (3).
imprimitivity of non-symmetric 3-schemes
scientific article; zbMATH DE number 847628