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A new Young wall realization of $B(λ)$ and $B(\infty)$

Zhaobing Fan, Shaolong Han, Seok-Jin Kang, Young Rock Kim

Abstract

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal $B(λ)$ for the quantum affine algebras of types $A^{(2)}_{2n}$, $D^{(2)}_{n+1}$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$, $B^{(1)}_n$ and $C^{(1)}_n$. We show that the crystal consisting of reduced Young walls is isomorphic to the crystal $B(λ)$. Moreover, we provide a new realization of the crystal $B(\infty)$ in terms of reduced virtual Young walls and reduced extended Young walls.

A new Young wall realization of $B(λ)$ and $B(\infty)$

Abstract

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal for the quantum affine algebras of types , , , , and . We show that the crystal consisting of reduced Young walls is isomorphic to the crystal . Moreover, we provide a new realization of the crystal in terms of reduced virtual Young walls and reduced extended Young walls.
Paper Structure (10 sections, 6 theorems, 62 equations)

This paper contains 10 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Quantum affine algebras and perfect crystals
  3. Higher level Young columns
  4. Level-$l$ stacking patterns and Young columns
  5. Level-$l$ reduced Young columns
  6. The crystal structure on Young columns
  7. Higher level Young walls
  8. Ground-state columns and ground-state walls
  9. New Young wall realization of the crystal $B(\lambda)$
  10. New Young wall realization of the cryatal $B(\infty)$

Key Result

Proposition 2.5

For each $\lambda \in P^+_{\text{cl}}$, there exists a $U_q'(\mathfrak g)$-crystal isomorphism

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: KMN1KMN2
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.10
  • proof
  • ...and 7 more