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Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Duncan Laurie

Abstract

We construct Young wall models for the crystal bases of level $1$ irreducible highest weight representations and Fock space representations of quantum affine algebras in types $E_{6}^{(1)}$, $E_{7}^{(1)}$ and $E_{8}^{(1)}$. In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Abstract

We construct Young wall models for the crystal bases of level irreducible highest weight representations and Fock space representations of quantum affine algebras in types , and . In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.
Paper Structure (18 sections, 27 theorems, 32 equations, 17 figures, 3 tables)

This paper contains 18 sections, 27 theorems, 32 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations of quantum affine algebras
  4. Crystals of quantum affine algebras
  5. Energy functions
  6. Fock space representations
  7. Level 1 perfect crystals in type E
  8. The level 1 perfect crystals of Benkart-Frenkel-Kang-Lee
  9. Energy functions
  10. Young column realizations
  11. Types E6(1) and E7(1)
  12. Type E8(1)
  13. Level 1 irreducible highest weight crystals in type E
  14. Fock space crystals in type E
  15. Analysing the structure of Young walls in Z(Lambda0)
  16. ...and 3 more sections

Key Result

Theorem 2.4

Kashiwara91 For each $\lambda\in P^{+}$ the lattice $L(\lambda)$ is the smallest $A$-submodule of $V(\lambda)$ containing $u_{\lambda}$, and $B(\lambda)$ is the set of all non-zero vectors in $L(\lambda)/qL(\lambda)$ of the form $\Tilde{f}_{i_{\ell}}\dots\Tilde{f}_{i_{1}}u_{\lambda}$.

Figures (17)

  • Figure 1: The type $E$ untwisted affine Dynkin diagrams. Black labels are vertex numbers, and blue labels are the numerical values $a_{i} = a_{i}^{\vee}$
  • Figure 2: The crystal graph of $B_{6}$
  • Figure 3: The crystal graph of $B_{7}$
  • Figure 4: Examples of displaying cut unit cubes
  • Figure 5: Young column patterns for types $E_{6}^{(1)}$ and $E_{7}^{(1)}$
  • ...and 12 more figures

Theorems & Definitions (75)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 65 more