Markov traces on degenerate cyclotomic Hecke algebras

TL;DR

This paper constructs Markov traces on degenerate cyclotomic Hecke algebras, enabling a canonical symmetrizing form and connecting the Brudan--Kleshchev trace as a specialization, advancing algebraic understanding.

Contribution

It introduces and constructs Markov traces on degenerate cyclotomic Hecke algebras, providing a new canonical symmetrizing form and linking existing traces as special cases.

Findings

Defined and constructed Markov traces on $H_n(oldsymbol{u})$

Established a canonical symmetrizing form on $H_n(oldsymbol{u})$

Connected Brudan--Kleshchev trace to Markov traces as a specialization

Abstract

Let $H_n(\boldsymbol{u})$ be the degenerate cyclotomic Hecke algebra with parameter $\boldsymbol{u}=(u_1, \ldots, u_m)$ over $\mathbb{C}(\boldsymbol{u})$. We define and construct the (non-)normalized Markov traces on the sequence $\{H_n(\boldsymbol{u})\}_{n=1}^{\infty}$. This allows us to provide a canonical symmetrizing form on $H_n(\boldsymbol{u})$ and show that the Brudan--Kleshchev trace on $H_n(\boldsymbol{u})$ is a specialization of the non-normalized Markov traces.