Representations of Temperley--Lieb Algebras

We define a commuting family of operators $T_0,T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_n$ 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.