Non-minimality of certain irregular coherent preminimal affinizations

Abstract

Let g be a finite-dimensional simple Lie algebra of type D or E and λ be a dominant integral weight whose support bounds the subdiagram of type D4. We study certain quantum affinizations of the simple g-module of highest weight λ which we term preminimal affinizations of order two (this is the maximal order for such λ). This class can be split in two: the coherent and the incoherent affinizations. If λ is regular, Chari and Pressley proved that the associated minimal affinizations belong to one of the three equivalent classes of coherent preminimal affinizations. In this paper we show that, if λ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if g is of type D4, this completes the classification of minimal affinizations for type D4 by giving a negative answer to a conjecture of Chari-Pressley stating that the coherent and the incoherent affinizations were equivalent in type D4 (this corrects the opposite claim made by the first author in a previous publication).