Representations of Graph Temperley-Lieb Algebras

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Abstract
We define graph Temperley-Lieb algebras, which are certain generalisations of Temperley-Lieb algebras including those appropriate for building the transfer matrices for lattice models in arbitrary dimensions. We construct various representations for these algebras, which are parameterised by a scalar Q . In particular we give the generically irreducible representations associated with statistical mechanical models. These representations are well defined for all Q , whereas the representations coming directly from physical models are typically only defined for certain values of Q . We also give representations with bases derived from the partitions of n distinguishable objects ( n is the number of nodes in the graph). We show how to compute the dimensions of these epresentations by a diagrammatic technique.

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