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.
Categories: math.RT