Crystals and quantum twist automorphisms

Abstract

Let $\eta_w$ be the quantum twist automorphism for the quantum unipotent coordinate ring $\mathrm{A}_q(\mathfrak{n}(w))$ introduced by Kimura and Oya. In this paper, we study the quantum twist automorphism $\eta_w$ in the viewpoint of the crystal bases theory and provide a crystal-theoretic description of $\eta_w$. In the case of the $*$-twisted minuscule crystals of classical finite types, we provide a combinatorial description of $\eta_w$ in terms of (shifted) Young diagrams. We further investigate the periodicity of $\eta_w$ up to a multiple of frozen variables in various setting.