Abstract
The Lie algebraic structures of the S-matrices for the affine Toda field theories based on the dual pairs ( X N (1), Y M ( l) ) are discussed. For the non-simply laced horizontal subalgebra X N and the simply laced horizontal subalgebra Y M , we introduce a “ q-deformation” of a Coxeter element and a “ p-deformation” of a twisted Coxeter element respectively. Using these deformed objects, expressions for the generating function of the multiplicities of the building block of the S-matrices are obtained. The relation between the deformed version of Dorey's rule and the generalized Dorey's rule due to Chari and Pressley are discussed.
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