Davenport-Hasse theorem and cyclotomic association schemes

Definition. Let q be a prime power and e be a divisor of q − 1. Fix a generator α of the multiplicative group of GF (q). Then 〈α〉 is a subgroup of index e and its cosets are 〈α〉α, i = 0, . . . , e− 1. Define R0 = {(x, x)|x ∈ GF (q)} Ri = {(x, y)|x, y ∈ GF (q), x− y ∈ 〈αe〉αi−1}, (1 ≤ i ≤ e) R = {Ri|0 ≤ i ≤ e} Then (GF (q),R) forms an association scheme and is called the cyclotomic scheme of class e on GF (q). The character table of the cyclotomic scheme is the first eigenmatrix of the association scheme. We will not give a formal definition here, instead, we show how to construct the character table of cyclotomic scheme from the character table of elementary abelian group. For a discussion of association scheme in general and the definition of the character table, see [2]. Consider the character table T of the additive group of F = GF (q):