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A harmonic polynomial of degree i is defined to be an element of the column space of E i . A polynomial function of degree i is a linear combination of harmonic polynomials with degree at most i . The previous result implies that if f is a polynomial with degree 1 and g is a polynomial with degree i , then f ◦ g has degree at most i + 1. Note that f ◦ g is just the usual product of functions. 5 The Frame Quotient Let A be an association scheme with d classes on the vertex set v. Let e u be the characteristic vector of the vertex u and let H be the matrix H := (A 0 e u , A 1 e u , · · · , A d e u ).

The association scheme of a strongly regular graph. The Schur idempotents of A ⊗ A are the nine matrices I, I ⊗ A1, A1 ⊗ I , I ⊗ A2, A2 ⊗ I , A1 ⊗ A2, A2 ⊗ A1, A1 ⊗ A1, A2 ⊗ A2. The Schur idempotents of H (2, A ) are I, I ⊗ A1 + A1 ⊗ I , I ⊗ A2 + A2 ⊗ I , A1 ⊗ A2 + A2 ⊗ A1, A1 ⊗ A1, A2 ⊗ A2. 5 A Tensor Identity We use A ⊗ B to denote the Kronecker product of two matrices A and B . We offer a more exalted version of Seidel’s identity, due to Koppinen. 1 Theorem. Let A be an association scheme with d classes.

1 An Inner Product There is one important property of Bose-Mesner algebras still to be discussed. If M , N ∈ Matm×n (C), we define 〈M , N 〉 := tr(M ∗ N ). As is well known, this is a complex inner product on Matm×n (C). Note that 〈N , M 〉 = 〈M , N 〉. If sum(M ) denotes the sum of the entries of the matrix M , then tr(M ∗ N ) = sum(M ◦ N ) and therefore 〈M , N 〉 = sum(M ◦ N ). It follows that the Bose-Mesner algebra of an association scheme A is an inner product space. If A = {A 0 , . . , A d } 23 24 CHAPTER 3.

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