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

Taehyeok Heo

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.

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

Abstract

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

Paper Structure

This paper contains 20 sections, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crystals
  4. The polyhedral realization
  5. The PBW crystal
  6. The extended crystal
  7. A combinatorial description of the bicrystal $\mathcal{B}(\infty)$
  8. Settings
  9. A description on the bicrystal structure of $B(\infty)$
  10. A sliding diamond rule
  11. A combinatorial description of the extended crystal
  12. A combinatorial description of the extended crystal
  13. An extended sliding diamond rule for type $A_n$
  14. An application to representation theory: categorical crystal
  15. The categorical crystal
  16. ...and 5 more sections

Theorems & Definitions (9)

  • proof
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  • proof
  • proof : Proof of Proposition \ref{['prop:polyhed star crystal']}
  • proof
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  • proof