Calibrated representations of two boundary Temperley-Lieb algebras

The two boundary Temperley-Lieb algebra $TL_k$ arises in the transfer matrix formulation of lattice models in Statistical Mechanics, in particular in the introduction of integrable boundary terms to the six-vertex model. In this paper, we classify and study the calibrated representations---those for which all the Murphy elements (integrals) are simultaneously diagonalizable---which, in turn, corresponds to diagonalizing the transfer matrix in the associated model. Our approach is founded upon the realization of $TL_k$ as a quotient of the type $C_k$ affine Hecke algebra $H_k$. In previous work, we studied this Hecke algebra via its presentation by braid diagrams, tensor space operators, and related combinatorial constructions. That work is directly applied herein to give a combinatorial classification and construction of all irreducible calibrated $TL_k$-modules and explain how these modules also arise from a Schur-Weyl duality with the quantum group $U_q\mathfrak{gl}_2$.