The two-boundary Temperley-Lieb algebra

Abstract

We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of the centre. For generic values of this parameter there is a unique representation of dimension 2^N and we show that it is isomorphic to a tensor space representation. We construct a basis in which the Gram matrix is diagonal and use this to discuss the irreducibility of this representation.

45 pages Latex, 21 eps figures, revised version

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Cited by in corpus (44)

• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
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• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Boundary algebras and Kac modules for logarithmic minimal models
• Combinatorial aspects of boundary loop models
• Koornwinder polynomials and the XXZ spin chain
• Exact finite size groundstate of the O(n=1) loop model with open boundaries
• Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Finite-size corrections for universal boundary entropy in bond percolation
• On the non-generic representation theory of the symplectic blob algebra
• Two boundary Hecke Algebras and combinatorics of type C
• Quantum spin chains from Onsager algebras and reflection $K$-matrices
• On boundary fusion and functional relations in the Baxterized affine Hecke algebra
• Can a spin chain relate combinatorics to number theory?
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Shor-Movassagh chain leads to unusual integrable model
• Tilting Modules for the Symplectic Blob Algebra
• A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain
• Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices
• Degenerate two-boundary centralizer algebras
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Two-point boundary correlation functions of dense loop models
• Connection problems for quantum affine KZ equations and integrable lattice models
• Calibrated representations of two boundary Temperley-Lieb algebras
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• A Tensor Space Representation of the Symplectic Blob Algebra
• Finite size lattice results for the two-boundary Temperley--Lieb loop model
• On quasi-heredity and cell module homomorphisms in the symplectic blob algebra
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian
• Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}{\mathfrak{sl}}_2$ Symmetry
• Logarithmic Minimal Models with Robin Boundary Conditions
• Diagram calculus for a type affine $C$ Temperley--Lieb algebra, I
• Non-Interacting Motzkin Chain - Periodic Boundary Conditions
• Handlebody diagram algebras
• Decomposition matrices and blocks for the symplectic blob algebra over the complex field
• The ghost algebra and the dilute ghost algebra
• Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Junction type representations of the Temperley-Lieb algebra and associated symmetries
• The planar parafermion algebra: The ${\mathbb{Z}}_N$ clock model and the coupled Temperley-Lieb algebra
• The nil Temperley--Lieb algebra of type affine C
• Exotic Springer fibers for orbits corresponding to one-row bipartitions