Math Problem Statement

Suppose A is a scheme with d classes. Since C[A ] is closed under multiplica- tion, there are constants pi,j(k) such that Ai Aj = X d k=0 pi,j(k)Ak. We call these the intersection numbers of the scheme. We see that pi,j(k)Ak = Ak ◦ (Ai Aj), from which it follows that the intersection numbers are non-negative integers. We see also that pi,j(k) = sum(Ak ◦ (Ai Aj)) v vk

tr(A T k Ai Aj) v vk . (2.3.1) We define the intersection matrices B0,...,Bd by (Bi)j,k := pi,j(k). If π denotes the relation partition of V (A ) with respect to v, then Bi = A/π.

Solution

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Math Problem Analysis

Mathematical Concepts

Algebraic Combinatorics
Association Schemes
Matrix Algebra
Adjacency Matrices

Formulas

Ai Aj = ∑_(k=0)^d pi,j(k) Ak
pi,j(k) = (sum(Ak ◦ (Ai Aj))) / (v vk) = tr(Ak^T Ai Aj) / (v vk)

Theorems

Intersection Numbers Theorem in Association Schemes

Suitable Grade Level

Advanced Undergraduate or Graduate Level