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Extra resources for Association Schemes [Lecture notes]

Sample text

Let P be the eigenmatrix of an association scheme A with d classes. If σ is a tight partition of the columns of P , then P S = RP 1 where R is the characteristic matrix of σ∗ . In this case P is invertible, and so P 1 is invertible. If Q is the dual eigenmatrix of A , then QP = v I and we have vSP 1−1 = QR. This implies that σ∗ is a tight partition of the columns of Q, with induced partition equal to σ. If the Schur idempotents associated with the cells of σ form a subscheme of A , then P 1 is the matrix of eigenvalues of the subscheme.

If i ∈ R then x T A i x = 0 and, if i ∉ R ∪ {0}, then a i x T A i x ≤ 0. 3. LINEAR PROGRAMMING 27 and xT N x = b j x T E j x ≥ b0 x T E 0 x ≥ j b0 2 |S| . v Hence |S| ≤ v a0 tr(N ) =v . b0 sum(N ) Thus we have the following. 2 Theorem. Let A be an association scheme with d classes and let S be an R-coclique in it. Then tr(N ) |S| ≤ min v N sum(N ) where N runs over the set of positive semidefinite matrices in C[A ] such that N ◦ A i ≤ 0 if i ∉ R ∪ {0}. From this theorem we also see that v sum(N ) ≥ max N |S| tr(N ) where N runs over the set of positive semidefinite matrices in C[A ] such that N ◦ A i ≤ 0.

In this case P is invertible, and so P 1 is invertible. If Q is the dual eigenmatrix of A , then QP = v I and we have vSP 1−1 = QR. This implies that σ∗ is a tight partition of the columns of Q, with induced partition equal to σ. If the Schur idempotents associated with the cells of σ form a subscheme of A , then P 1 is the matrix of eigenvalues of the subscheme. 3 Primitivity An association scheme with d classes is primitive if each of its graphs X 1 , . . , X d is connected. ) An association scheme that is not primitive is imprimitive.