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

Abstract

This paper studies the seminormal bases {fst},{fst}, and the dual seminormal bases {gst},{gst} of the non-degenerate and the degenerate cyclotomic Hecke algebras Hℓ,n of type G(ℓ,1,n). We present some explicit formulae for the constants αst:=gst/fst∈K×, ast:=gst/fst∈K× in terms of the γ-coefficients {γu,γu′} and the r-coefficients {ru,ru′} of Hℓ,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 Hℓ,n−1 as a linear combination of the seminormal bases of Hℓ,n under the natural inclusion Hℓ,n−1↪Hℓ,n.