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

Abstract

We define a unified trace form $\tau$ on the cyclotomic Hecke algebras ${\mathcal{H}}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $\xi \ne 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for ${\mathcal{H}}_{n,K}$ with respect to the form. We also construct an explicit basis for the cocenter (i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field $K$, the Hecke parameter $\xi$ and the cyclotomic parameters $Q_1,\cdots ,Q_{\ell }$.

Correct two typos in Lemma 4.4 and Theorem 4.7