Maximizing numerical radii of weighted shifts under weight permutations

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