Crystals and quantum twist automorphisms

TL;DR

This paper explores the quantum twist automorphism in quantum unipotent coordinate rings, providing a crystal-theoretic and combinatorial description, especially for minuscule crystals, and investigates its periodicity.

Contribution

It offers a crystal-theoretic framework for understanding quantum twist automorphisms and describes them combinatorially for classical types, advancing the understanding of their structure.

Findings

Crystal-theoretic description of quantum twist automorphisms.

Combinatorial description using Young diagrams for classical types.

Analysis of the periodicity of automorphisms in various settings.

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.