HALF POTENTIAL ON GEOMETRIC CRYSTALS AND CONNECTEDNESS OF CELLULAR CRYSTALS

Abstract

For any simply connected, simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell \( \left(=B{\overline{w}}_0:= {B}^{-}\cap U{\overline{w}}_0U\right) \) is isomorphic to the Langlands dual crystal B(∞) of the nilpotent-half subalgebra of quantum group. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.