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

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