On the Fermionic Formula and the Kirillov-reshetikhin Conjecture

The irreducible finite-dimensional representations of quantum affine algebras Uq(ĝ) have been studied from various viewpoints, [AK], [CP1], [CP3], [C], [CP4], [FR], [FM], [KR], [KS]. These representations decompose as a direct sum of irreducible representations of the quantized eneveloping algebra Uq(g) associated to the underlying finite-dimensional simple Lie algebra g. But, except for a few special cases, little is known about the isotypical components occuring in the decomposition. However, for a certain class of modules (namely the one associated in a canonical way to a multiple of a fundamental weight of g), there is a conjecture due to Kirillov and Reshetikhin [KR] for Yangians that describes the g-isotypical components. (Actually, the fermionic formula given in [KR] applies to the tensor products of these representations, but we shall not deal with that case here.) A combinatorial interpretation of their conjecture in the case of the multiple of a fundamental representation was given by Kleber, [Kl], when g is simply-laced (see also [HKOTY]). It is the purpose of this paper to prove the conjecture for the quantum affine algebras associated to the simply–laced algebras, using Kleber’s interpretation. We now describe the conjecture and the results more explicitly. Let λ1, λ2, · · · , λn be a set of fundamental weights for g and, for any dominant integral weight μ, let V (μ) denote the irreducible Uq(g)-module with highest weight μ. If g is of type An, the conjecture just says that the modules V (mλi) admit a Uq(Ân)-module structure. But this is well-known: in fact, it is known, by using the evaluation homomorphism [CP2], that any Uq(An)-module admits the structure of a Uq(Ân)module. In the case when g is of type Dn, the conjecture (as interpreted by Kleber) asserts the existence of a Uq(ĝ)-module W (mλi), for all m ≥ 0 and with λi not corresponding to one of the spin representations, such that, as U(g)-modules, W (mλi) = ⊕