Semisimplicity of adjacency algebras of association schemes (Q1972028)

Let \(Y=(X,\{R_i\}_{0\leq i\leq d})\) be an association scheme and \(n=|X|\). \(R\) be a commutative ring with 1. Define \(RY=\bigoplus_{i=0}^d Ra_i\) with the multiplication \(a_ia_j=\sum_k p_{ij}^ka_k\) and call \(RY\) the adjacency algebra of \(Y\) over \(R\). We call \(\{a_i\}\) the standard basis of \(Y\). We can define an \(RY\)-module \(RX\), which is an \(R\)-free module of rank \(n\) and indexed by \(X\). Let \(\mathbb{C}\) be the complex number field. Then \({\mathbb{C}}X=\bigoplus_{i=1}^r m_iS_i\), where \(S_i\) is an irreducible \({\mathbb{C}}Y\)-module. Put \(f_i= \dim S_i\). We define a frame number \({\mathcal F}(Y)\) by \[ {\mathcal F}(Y)=n^{d+1}\frac{\prod_{i=0}^d v_i}{\prod_{i=1}^r m_i^{f_i^2}}. \] The author investigates the semisimplicity of an adjacency algebra of association schemes over positive characteristic fields. Theorem 4.2. Let \(k\) be a field of characteristic \(p\). Then \(kY\) is semisimple if and only if the frame number \({\mathcal F}(Y)\) is not divided by \(p\).