Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

We define a unified trace form τ on the cyclotomic Hecke algebras H_n,K of type A, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter ξ≠1) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct an explicit pair of dual bases for H_n,K with respect to the form. We also construct an explicit basis for the cocenter of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field K, the Hecke parameter ξ and the cyclotomic parameters Q₁,⋯,Q_ℓ.