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.

arXiv comment42 pages

References in corpus (10)

• Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
• Cluster algebras and quantum affine algebras
• Crystals from categorified quantum groups
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras
• Polyhedral Realizations of Crystal Bases for Quantum Algebras of Finite Types
• A rigged configuration model for $B(\infty )$
• Combinatorial descriptions of the crystal structure on certain PBW bases
• Rigged configurations for all symmetrizable types
• Rigged configurations and the $\ast$-involution
• Alcove path model for $B(\infty )$