Combinatorial aspects of boundary loop models

Abstract

We discuss in this paper combinatorial aspects of boundary loop models, that ismodels of self-avoiding loops on a strip where loops get different weights dependingon whether they touch the left, the right, both or no boundary. These modelsare described algebraically by a generalization of the Temperley–Lieb algebra,dubbed the two-boundary TL algebra. We give results for the dimensions of TLrepresentations and the corresponding degeneracies in the partition functions. We interpretthese results in terms of fusion and in the light of the recently uncovered large symmetry present in loop models, paving the way for the analysis of the conformalfield theory properties. Finally, we propose conjectures for determinants of Gram matricesin all cases, including the two-boundary one, which has recently been discussed by de Gierand Nichols.