Association Schemes [Lecture notes] by Chris Godsil

By Chris Godsil

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Do quotients schemes a là Godsil and Martin Chapter 6 Translation Schemes Suppose Γ is an abelian group of order v. The conjugacy class scheme on Γ is a scheme with v − 1 classes, and each minimal Schur idempotent is a permutation matrix. Many interesting schemes arise as subschemes of these; they are known as translation schemes. 1 Characters Let Γ be a finite abelian group. A character of Γ is a homomorphism from Γ into the multiplicative group formed by the non-zero complex numbers. The set of all characters of Γ is denoted by Γ, and is called the character group of Γ.

A d and vertex set V . 5. A TENSOR IDENTITY 39 d + 1 whose i -th entry h i (u, v) is the number of coordinates j such that u j and v j are i -related. The entries of h(u, v) sum to n; conversely any non-negative vector of length n whose entries sum to n is equal to h(u, v) for some u and v. If α is a non-negative vector of length d + 1 and 1T α = n, define A α to be the 01-matrix with rows and columns indexed by V n and with (A α )u,v = 1 if and only if h(u, v) = α. This set of matrices forms the k-th symmetric power of A .

X v be a basis for Fv such that x 1 , . . , x k are the columns of H in order. 2. SUBSCHEMES AND PARTITIONS 45 A has the form B 0 ∗ A1 where A 1 is square. Therefore det(t I − A) = det(t I − B ) det(t I − A 1 ). We give one of the standard applications of this result. Let X be a graph. A perfect 1-code in X is a subset C of V (X ) such that: (a) Any two vertices in C are at distance at least three. (b) Any vertex not in C is adjacent to exactly one vertex in C . Suppose C is a perfect 1-code in X , and let π be the partition with cells C 1 = C and C 2 = V (X )\C .

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