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Abstract: We discuss in this paper combinatorial aspects of boundary loop models, thatis models of self-avoiding loops on a strip where loops get different weightsdepending on whether they touch the left, the right, both or no boundary. Thesemodels are described algebraically by a generalization of the Temperley-Liebalgebra, dubbed the two-boundary TL algebra. We give results for the dimensionsof TL representations and the corresponding degeneracies in the partitionfunctions. We interpret these results in terms of fusion and in the light ofthe recently uncovered A n large symmetry present in loop models, paving theway for the analysis of the conformal field theory properties. Finally, wepropose conjectures for determinants of Gram matrices in all cases, includingthe two-boundary one, which has recently been discussed by de Gier and Nichols.

Autor: Jesper Lykke Jacobsen LPTMS, SPhT, Hubert Saleur SPhT


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