Maximizing numerical radii of weighted shifts under weight permutations

Let wi∈C (1≤i≤n) and l∈Sn, the symmetric group of all permutations of 1,2,…,n. Suppose Al is the weighted cyclic matrix Al≡(0wl(1)0⋱⋱wl(n−1)wl(n)0) and w(Al) denotes its numerical radius. We characterize those ζ∈Sn which satisfy w(Aζ)=maxl∈Snw(Al). The characterizations for unilateral and bilateral weighted (backward) shifts are also obtained.