  In this paper we study minimal affinizations of representations of quantum
groups (generalizations of Kirillov-Reshetikhin modules of quantum affine
algebras introduced by Chari). We prove that all minimal affinizations in types
A, B, G are special in the sense of monomials. Although this property is not
satisfied in general, we also prove an analog property for a large class of
minimal affinization in types C, D, F. As an application, the Frenkel-Mukhin
algorithm works for these modules. For minimal affinizations of type A, B we
prove the thin property (the l-weight spaces are of dimension 1) and a
conjecture of Nakai-Nakanishi (already known for type A). The proof of the
special property is extended uniformly for more general quantum affinizations
of quantum Kac-Moody algebras.
