Geometric Crystals on Flag Varieties and Unipotent Subgroups of Classical Groups

TL;DR

This paper proves that for classical simple algebraic groups, the geometric crystal structure on the flag variety is isomorphic to that on the unipotent subgroup, confirming a conjecture in the field.

Contribution

It establishes an explicit isomorphism between geometric crystals on flag varieties and unipotent subgroups for classical groups, confirming a key conjecture.

Findings

Confirmed the conjecture for classical groups

Established isomorphism between geometric crystals on flag varieties and unipotent subgroups

Advances understanding of geometric crystal structures in algebraic groups

Abstract

For a classical simple algebraic group $G$ we obtain the affirmative answer for the conjecture in [8] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the unipotent subgroup $U^-$.