Standard symmetrizing forms and Murphy bases of the cyclotomic Hecke algebras

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 $G(\ell,1,n)$.