Vanishing conditions on parameters for \(P(Q)\)-polynomial association schemes (Q5943884)

Let \(X\) be a finite nonempty set of size \(n\). Let \(\mathcal M\) be a Bose-Mesner algebra on \(X\). For fixed \(p\in X\) and for \(A\in {\mathcal M}\) let \(\rho(A)\) denote the diagonal matrix with \((x,x)\)-entry \(\rho(A)(x,x)=A(p,x)\). Set \({\mathcal M}^*=\rho({\mathcal M})\). Then the Terwilliger algebra \({\mathcal T}={\mathcal T}({\mathcal M},p)\) associated with \(\mathcal M\) and \(p\) is the subalgebra of \(M_X\), generated by \({\mathcal M}\cup {\mathcal M}^*\). Let \(E_0,\dots{},E_d\) (\(E_0^*,\dots{},E_d^*\)) be the (dual) primitive idempotents of \(\mathcal M\) (\({\mathcal M}^*)\). An irreducible \(\mathcal T\)-module \(W\) is said to be (dual) \(i\)-thin if \(\dim E_i^*(W)\leq 1\) \((\dim E_i(W)\leq 1)\) for some \(0\leq i\leq d\). NEWLINENEWLINENEWLINETheorem 1. Let \(Y\) denote a \(d\)-class symmetric association scheme, with \(d\geq 3\). Suppose \(A_0,\dots{},A_d\) is a \(P\)-polynomial structure for \(Y\), with intersection numbers \(p^h_{ij}\), and suppose for \(0\leq i< j\leq d\) and \(0<i+j\leq d\), \(Y\) is \(i\)-thin with respect to at least one vertex. Then \(a_{j-1}=0\), and \(p^{i+j-2}_{i,j-1}=0\) implies \(a_{i+j-1}=0\). NEWLINENEWLINENEWLINETheorem 2. Let \(Y\) denote a \(d\)-class symmetric association scheme, with \(d\geq 3\). Suppose \(E_0,\dots{},E_d\) is a \(Q\)-polynomial structure for \(Y\), with Krein parameters \(q^h_{ij}\), and suppose for \(0\leq i< j\leq d\) and \(0<i+j\leq d\), \(Y\) is dual \(i\)-thin with respect to at least one vertex. Then \(a_{j-1}^*=0\), and \(q^{i+j-2}_{i,j-1}=0\) implies \(a^*_{i+j-1}=0\).