A new description of the bicrystal $B(\infty)$ and the extended crystal

Abstract

We define new crystal maps on $B(\infty)$ using its polyhedral realization, and show that the crystal $B(\infty)$ equipped with the new crystal maps is isomorphic to Kashiwara's $B(\infty)$ as bicrystals. In addition, we combinatorially describe the bicrystal structure of $B(\infty)$, which is called a sliding diamond rule. Using the bicrystal structure on $B(\infty)$, we define the extended crystal and show that it is isomorphic to the extended crystal introduced by Kashiwara and Park.