Tangles in affine Hecke algebras

TL;DR

This paper demonstrates that affine Hecke algebras of type A can be represented using framed oriented tangle diagrams in the annulus, extending the algebra's diagrammatic and coefficient framework.

Contribution

It introduces a tangle diagrammatic model for affine Hecke algebras using Homfly skein relations, broadening the diagrammatic tools and understanding of these algebras.

Findings

Isomorphism between tangle diagrams and affine Hecke algebra

Enhanced diagrammatic representation including closed curves

Clear visualizations of central elements

Abstract

The affine Hecke algebra $\dot H_n$ of type $A$ is often presented as a quotient of the braid algebra of $n$-braids in the annulus. This leads to diagrammatic representations in terms of braids in the annulus, subject to a quadratic relation for the simple Artin braids, as in the description by Graham and Lehrer in \cite{GL03}. I show here that the use of more general framed oriented $n$-tangle diagrams in the annulus, subject to the Homfly skein relations, produces an algebra which is isomorphic to $\dot H_n$ with an extended ring of coefficients. This setting allows the use of some attractive diagrams for elements of $\dot H_n$, using closed curves as well as braids, and gives neat pictures for its central elements.