Representations of Temperley--Lieb Algebras

TL;DR

This paper introduces a semi-normal form and branching law for Temperley--Lieb algebras of type A, using a Murphy basis analogue to analyze eigenvalues and Gram determinants.

Contribution

It develops a semi-normal form, branching law, and explicit Gram determinant formulas for Temperley--Lieb algebras of type A using a Murphy basis analogue.

Findings

Eigenvalues of operators on cell modules are characterized.

A semi-normal form for Temperley--Lieb algebras is established.

Explicit Gram determinant formulas are derived.

Abstract

We define a commuting family of operators $T_0,T_1,...,T_n$ in the Temperley--Lieb algebra $\mathcal{A}_n(x)$ of type $A_{n-1}$. Using an appropriate analogue to Murphy basis of the Iwahori--Hecke algebra of the symmetric group, we describe the eigenvalues arising from the triangular action of the said operators on the cell modules of $\mathcal{A}_n(x)$. These results are used to provide the Temperley--Lieb algebras of type $A_{n-1}$ with a semi--normal form, together with a branching law, and explicit formulae for associated Gram determinants.