Existence of Kirillov-Reshetikhin crystals for near adjoint nodes in exceptional types

Abstract

We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang et.al. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules.

arXiv comment39 pages. In version 1 the main theorem is proved in type $E_6^{(1)}$ only, but in version 2 and 3 we generalize the result in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$. In version 3, some minor corrections are made

References in corpus (6)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• On Level Zero Representations of Quantized Affine Algebras
• Proof of the combinatorial Kirillov-Reshetikhin conjecture
• Finite Dimensional Representations of Quantum Affine Algebras
• Existence of Kirillov--Reshetikhin crystals for multiplicity free nodes
• Uniform description of the rigged configuration bijection

Cited by in corpus (4)

• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with a minuscule node $r$