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

Abstract: 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 $H_{\ell,n-1}\hookrightarrow H_{\ell,n}$.