Non-Minimality of Certain Irregular Coherent Preminimal Affinizations

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra of type $D$ or $E$ and $\lambda$ be a dominant integral weight whose support bounds the subdiagram of type $D_4$. We study certain quantum affinizations of the simple $\mathfrak g$-module of highest weight $\lambda$ which we term preminimal affinizations of order two (this is the maximal order for such $\lambda$). This class can be split in two: the coherent and the incoherent affinizations. If $\lambda$ 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 $\lambda$ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if $\mathfrak g$ is of type $D_4$, this completes the classification of minimal affinizations for type $D_4$ by giving a negative answer to a conjecture of Chari-Pressley stating that the coherent and the incoherent affinizations were equivalent in type $D_4$ (this corrects the opposite claim made by the first author in a previous publication).