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

TL;DR

This paper introduces a unified trace form for cyclotomic Hecke algebras of type A, constructs dual bases using seminormal theory, and explicitly describes the cocenter of cyclotomic Schur algebras, revealing its dimension independence.

Contribution

It provides a unified trace form applicable to various cyclotomic Hecke algebras and explicitly constructs a basis for the cocenter of cyclotomic Schur algebras.

Findings

Unified trace form for cyclotomic Hecke algebras

Dual bases constructed via seminormal basis theory

Cocenter basis with dimension independent of parameters

Abstract

We define a unified trace form $\tau$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $\xi\neq 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ 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 $Q_1,\cdots,Q_\ell$.