Mathematics – Representation Theory
Scientific paper
2007-03-12
Journal of Algebra 321 (2009), 1132-1167
Mathematics
Representation Theory
45 pages Latex, 21 eps figures, revised version
Scientific paper
10.1016/j.jalgebra.2008.10.023
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.
de Gier Jan
Nichols Alexander
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