Frobenius map for the centers of Hecke algebras

TL;DR

The paper constructs a Frobenius map for the centers of Hecke algebras of symmetric groups, establishing an algebra isomorphism with symmetric functions and linking various bases.

Contribution

It introduces a new algebraic structure on the centers of Hecke algebras and connects them to symmetric functions through a Frobenius-type isomorphism.

Findings

Established an algebra isomorphism between centers and symmetric functions

Identified bases of centers with bases of symmetric functions

Provided explicit correspondences for distinguished bases

Abstract

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.