On the Markov trace for Temperley--Lieb algebras of type $E_n$

TL;DR

This paper establishes the uniqueness of a Markov trace on Temperley--Lieb quotients of Hecke algebras of type E_n, providing a diagrammatic computation method and applications to faithfulness and polynomial coefficients.

Contribution

It introduces a unique Markov trace for these algebras and details a diagrammatic approach for its computation, with applications to representation faithfulness and polynomial analysis.

Findings

Unique Markov trace exists for all n ≥ 6 in type E_n

Diagram calculus enables easy computation of the trace

Trace application to faithfulness and Kazhdan--Lusztig polynomials

Abstract

We show that there is a unique Markov trace on the tower of Temperley--Lieb type quotients of Hecke algebras of Coxeter type $E_n$ (for all $n \geq 6$). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan--Lusztig polynomials.