Symmetric Functions and Representations of Quantum Affine Algebras

Abstract

We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie algebras. One is defined via combinatorial properties and is easy to calculate; the other is closely related to the $q=1$ limit of the ``minimal affinization'' representations of quantum affine algebras. We conjecture that the two families are identical, and present supporting evidence and examples. In the special case of a highest weight that is a multiple of a fundamental weight, this reduces to a conjecture of Kirillov and Reshetikhin, recently proved by the first author.