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

TL;DR

This paper investigates the structure of seminormal and dual seminormal bases in cyclotomic Hecke algebras of type G(ℓ,1,n), providing explicit formulas and addressing questions about rationality of certain square roots.

Contribution

It introduces explicit formulas for constants relating seminormal and dual bases, and answers a question on the rationality of square roots of quotient products of gamma-coefficients.

Findings

Explicit formulas for basis constants in terms of gamma-coefficients.

Resolution of Mathas's question on rationality of certain square roots.

Formulas for expanding seminormal bases across algebra inclusions.

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