This paper studies the seminormal bases $\{ f_{ \mathfrak{ s}\mathfrak{ t}}\}$ and the dual seminormal bases $\{ g_{ \mathfrak{ s}\mathfrak{ t}}\}$ of the non-degenerate and the degenerate cyclotomic Hecke algebras ${ H}_{ \ell,n}$ of type $G(\ell,1,n)$. We present some explicit formulae for the constants $\alpha_{ \mathfrak{ s}\mathfrak{ t}}:=g_{ \mathfrak{ s}\mathfrak{ t}}/f_{ \mathfrak{ s}\mathfrak{ t}}\in K^\times$ in terms of the $\gamma$-coefficients $\{ \gamma_{ \mathfrak{ u}}, \gamma'_{ \mathfrak{ u}}\}$ of $H_{ \ell,n}$. In particular, we answer a question of Mathas on the rationality of square roots of some quotients of products of $\gamma$-coefficients. We obtain some explicit formulae for the expansion of each seminormal bases of $H_{ \ell,n-1}$ as a linear combination of the seminormal bases of $H_{ \ell,n}$ under the natural inclusion $H_{ \ell,n-1}\hookrightarrow H_{ \ell,n}$.
Categories: math.RT