Combinatorial aspects of boundary loop models

Abstract

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

v2: Added new figures 6, 9, 12, 13, 17. Further comparison with Ref. [7]

References in corpus (8)

• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Meanders and the Temperley-Lieb algebra
• Conformal boundary loop models
• One-boundary Temperley-Lieb algebras in the XXZ and loop models
• The two-boundary Temperley-Lieb algebra
• Character decomposition of Potts model partition functions. I. Cyclic geometry
• The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models

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• Boundary changing operators in the O(n) matrix model
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• Edge states and conformal boundary conditions in super spin chains and super sigma models
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• Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• Decomposition matrices and blocks for the symplectic blob algebra over the complex field