Markov traces on degenerate cyclotomic Hecke algebras

TL;DR

This work addresses the problem of defining and classifying Markov traces on the tower of degenerate cyclotomic Hecke algebras $H_n(\boldsymbol{u})$ to obtain canonical symmetrizing forms and connect with existing traces. It develops both normalized and non-normalized Markov traces using Lambropoulou's framework and an inductive basis built from Jucys–Murphy elements, establishing existence, uniqueness, and key structural properties. The normalized trace $\mathrm{tr}$ depends on parameters $z$ and $y_1,\dots,y_{m-1}$ and yields the Broué–Malle–Michel symmetrizing trace upon specialization, while the non-normalized trace $\mathrm{Tr}$ leads to Brundan–Kleshchev's trace as a specialization, tying together representation-theoretic and knot-theoretic perspectives. The results provide canonical trace forms on $H_n(\boldsymbol{u})$, enabling potential computation of Schur elements and linking to cyclotomic KLR algebras and related knot invariants in the solid torus.

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.

Theorems & Definitions (55)

• Definition 1.1
• Theorem 1.2
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Lemma 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• proof