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 $\{f_{\mathfrak{st}}\}$ and the dual seminormal bases $\{g_{\mathfrak{st}}\}$ 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 ${\alpha }_{\mathfrak{st}}:=g_{\mathfrak{st}}/f_{\mathfrak{st}}\in K^{\times }$ in terms of the $\gamma$-coefficients $\{{\gamma }_{\mathfrak{u}},{\gamma }_{\mathfrak{u}}^{\prime }\}$ 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 $H_{\ell ,n-1}\hookrightarrow H_{\ell ,n}$.