  We construct a new basis for a slim cyclotomic $q$-Schur algebra $\cysSr$ via
symmetric polynomials in Jucys--Murphy operators of the cyclotomic Hecke
algebra $\cysHr$. We show that this basis, labelled by matrices, is not the
double coset basis when $\cysHr$ is the Hecke algebra of a Coxeter group, but
coincides with the double coset basis for the corresponding group algebra, the
Hecke algebra at $q=1$. As further applications, we then discuss the cyclotomic
Schur--Weyl duality at the integral level. This also includes a category
equivalence and a classification of simple objects.
