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Soliton cellular automata for the affine general linear Lie superalgebra

Mitchell Ryan, Benjamin Solomon

Abstract

The box-ball system (BBS) is a cellular automaton that is an ultradiscrete analogue of the Korteweg--de Vries equation, a non-linear PDE used to model water waves. In 2001, Hikami and Inoue generalised the BBS to the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We further generalise the Hikami--Inoue BBS to column tableaux using the Kirillov--Reshetikhin crystals for $\hat{\mathfrak{gl}}{(m|n)}$ devised by Kwon and Okado (arXiv:1804.05456), where we find similar solitonic behaviour under certain conditions.

Soliton cellular automata for the affine general linear Lie superalgebra

Abstract

The box-ball system (BBS) is a cellular automaton that is an ultradiscrete analogue of the Korteweg--de Vries equation, a non-linear PDE used to model water waves. In 2001, Hikami and Inoue generalised the BBS to the general linear Lie superalgebra . We further generalise the Hikami--Inoue BBS to column tableaux using the Kirillov--Reshetikhin crystals for devised by Kwon and Okado (arXiv:1804.05456), where we find similar solitonic behaviour under certain conditions.
Paper Structure (24 sections, 24 theorems, 173 equations)

This paper contains 24 sections, 24 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The affine general linear Lie superalgebra
  4. Finite type crystals and tableaux
  5. Combinatorial R-matrix
  6. Energy function
  7. Super Box-Ball System
  8. Box-ball system definition
  9. Properties of the time evolution operator
  10. Solitons
  11. Properties of solitons and coupled solitons
  12. Solitons with speed equal to their length
  13. Scattering of two solitons
  14. Proof of Theorem \ref{['thm:speed']}
  15. Proof of Theorem \ref{['thm:scattering']}
  16. ...and 9 more sections

Key Result

Theorem 2.7

The combinatorial $R$-matrix maps $\mathcal{T}_1\otimes \mathcal{T}_2$ to $\widetilde{\mathcal{T}}_2\otimes \widetilde{\mathcal{T}}_1$ if and only if $\mathop{\mathrm{col}}\nolimits(\mathcal{T}_2)\rightarrow \mathcal{T}_1 = \mathop{\mathrm{col}}\nolimits(\widetilde{\mathcal{T}}_1)\rightarrow\widetil

Theorems & Definitions (83)

  • Example 1.1
  • Example 1.2
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7: KO18
  • Example 2.8
  • ...and 73 more