On the seminormal bases and dual seminormal bases of the cyclotomic Hecke algebras of type G(ℓ,1,n)

This paper studies the seminormal bases {f_st},{f_st}, and the dual seminormal bases {g_st},{g_st} of the non-degenerate and the degenerate cyclotomic Hecke algebras H_ℓ,n of type G(ℓ,1,n). We present some explicit formulae for the constants α_st:=g_st/f_st∈K^×, a_st:=g_st/f_st∈K^× in terms of the γ-coefficients {γ_u,γ_u^′} and the r-coefficients {r_u,r_u^′} of H_ℓ,n. In particular, we answer a question [20, Remark 3.6] of Mathas on the rationality of square roots of some quotients of products of γ-coefficients. We obtain some explicit formulae for the expansion of each seminormal bases of H_ℓ,n−1 as a linear combination of the seminormal bases of H_ℓ,n under the natural inclusion H_ℓ,n−1↪H_ℓ,n.