Crystals and quantum twist automorphisms

Abstract

Let $\hat{I}{\cdot }_w$ 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 $\hat{I}{\cdot }_w$ in the viewpoint of the crystal bases theory and provide a crystal-theoretic description of $\hat{I}{\cdot }_w$. In the case of the $\ast$-twisted minuscule crystals of classical finite types, we provide a combinatorial description of $\hat{I}{\cdot }_w$ in terms of (shifted) Young diagrams. We further investigate the periodicity of $\hat{I}{\cdot }_w$ up to a multiple of frozen variables in various setting.

51 pages