Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras
Zhekun He, Jun Hu, Huang Lin
TL;DR
This work constructs a unified trace form on cyclotomic Hecke algebras of type A that simultaneously generalizes the nondegenerate Malle–Mathas form and the degenerate Brundan–Kleshchev form. Using seminormal and cellular basis theory, it produces a pair of dual bases for the cyclotomic Hecke algebras and establishes a robust trace pairing, with implications for the cocenter and center via Schur–Weyl duality with cyclotomic Schur algebras. It then provides an explicit integral basis for the cocenter of the cyclotomic Schur algebra, showing the cocenter dimension is independent of the ground field and the Hecke/cyclotomic parameters, and demonstrates base-change stability. Collectively, these results illuminate center–cocenter duality, refine block-theoretic perspectives, and furnish concrete tools for the representation theory of cyclotomic Hecke and Schur algebras.
Abstract
We define a unified trace form $τ$ 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 $ξ\neq 1$) 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 $ξ$ and the cyclotomic parameters $Q_1,\cdots,Q_\ell$.