The multiplicities of a dual-thin \(Q\)-polynomial association scheme (Q5936674)

Let \(Y=(X,R)\) be an association scheme with \(D\) classes. If \(Y\) is \(P\)-polynomial with respect to an ordering \(R_0,\dots ,R_D\), then the corresponding sequence of valencies \(k_0,\dots ,k_D\) satisfies \(k_i\leq k_{i+1}\) and \(k_i\leq k_{D-i}\) for \(i<D/2\). Stanton made the following related conjecture: If \(Y\) is \(Q\)-polynomial with respect to an ordering \(E_0,\dots ,E_D\) of the primitive idempotents, then the corresponding multiplicities satisfy \(m_i\leq m_{i+1}\) and \(m_i\leq m_{D-i}\) for \(i<D/2\). NEWLINENEWLINENEWLINEFor \(x\in X\) let \(T=T(x)\) be the Terwilliger algebra with respect to \(x\). An irreducible \(T\)-module \(W\) is called dual-thin, if \(\dim(E_iW)\leq 1\) for \(0\leq i\leq D\). Finally \(Y\) is dual-thin if every irreducible \(T(x)\)-module \(W\) is dual-thin for every vertex \(x\in X\). In this paper it is proved that the above conjecture is true for dual-thin schemes.