Standard symmetrizing forms and Murphy bases of the cyclotomic Hecke algebras

TL;DR

This paper explicitly computes the standard symmetrizing form on Murphy basis elements of cyclotomic Hecke algebras of type G(ℓ,1,n), providing new formulas and relations between bases.

Contribution

It provides explicit calculations of the symmetrizing form on Murphy bases for both non-degenerate and degenerate cyclotomic Hecke algebras, and derives transition coefficient equations.

Findings

Explicit formulas for symmetrizing forms on Murphy bases.

Transition matrix relations between Murphy and seminormal bases.

Applications to algebraic structure and basis transformations.

Abstract

In this paper we compute the explicit value of the standard symmetrizing form of the cyclotomic Hecke algebra of type $G(\ell,1,n)$ on each Murphy basis element. We do this for both the non-degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$ and the degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$. As an application, we obtain some equations on the coefficients which occur in the transition matrix between the Murphy basis elements and the seminormal basis elements of the cyclotomic Hecke algebra of type $G(\ell,1,n)$.