The Structure of Finite Dimensional Affine Hecke Algebra Quotients and their Realization in 2D Lattice Models

Abstract

The affine Hecke algebra of type A N−1 is introduced and its finite dimensional representations are discussed. It is demonstrated, in a particular finite dimensional quotient, how generic and non-generic irreducible representations are obtained by diagonalizing the maximal commutative subalgebra. Examples of 2D models of statistical mechanics, in which the affine Hecke algebra is realized, are given. The twisted xxz quantum chain serves as an example of how a translation invariant model, which also gives a representation of the periodic Hecke algebra, can be analyzed using the representation theory of the affine Hecke algebra. It is shown how degeneracies in the spectrum of the Hamiltonian arise from the special structure of the non-generic irreducible representations. Finally, a more general relation between translation invariant lattice models and the affine Hecke algebra is proposed.