Unimodality of multiplicities in Q-polynomial association schemes

Determine whether the sequence of multiplicities of every Q-polynomial association scheme is unimodal.

Background

The paper studies multiplicities of Q-polynomial association schemes through the P–Q duality between valencies and multiplicities. It proves that the multiplicity sequence is log-concave for Q-polynomial schemes satisfying Property M, namely monotonicity of the dual intersection parameters b_i^* and c_i^*. Since log-concavity implies unimodality, this establishes unimodality only for that restricted class and leaves the general question for Q-polynomial association schemes unresolved.