Standard symmetrizing forms and Murphy bases of the cyclotomic Hecke algebras

In this paper we compute the explicit value of the standard symmetrizing form of the cyclotomic Hecke algebra of type $G(\mathrm{\ell },1,n)$ on each Murphy basis element. We do this for both the non-degenerate cyclotomic Hecke algebra of type $G(\mathrm{\ell },1,n)$ and the degenerate cyclotomic Hecke algebra of type $G(\mathrm{\ell },1,n)$. 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 $G(\mathrm{\ell },1,n)$.