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

A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations

“…It was conjectured by Hatayama et al [10] that for 1 ď r ď n and s P Z `, there exists a r,s P C ˆsuch that W prq s,ar,s has the crystal base introduced by Kashiwara [17]. The conjecture have been proved for r in the orbit of or adjacent to 0 in all affine types [19,20], all nonexceptional types [30], types G p1q 2 and D p3q 4 [28], and types E p1q 6,7,8 , F p1q 4 , E p2q 6 with near adjoint nodes [29]. Let B r,s be the crystal of the KR module, which is often called KR crystal for short.…”