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Soliton cellular automata associated with infinite reduced words. (English) Zbl 1451.14112

A soliton cellular automaton (SCA) is a cellular automaton which has solitonic solutions. In this paper the authors are interested in methods to construct SCA and to study their solutions applying combinatorics, representation theory and tropical geometry. They consider a family of cellular automata \(\Phi(n,k)\) associated with infinite reduced elements on the affine symmetric group \(\widehat{S}_n\), which is a tropicalization of the rational maps introduced in [M. Glick and P. Pylyavskyy, Transform. Groups 24, No. 1, 31–66 (2019; Zbl 1516.39007)]. The authors study the soliton solutions for \(F(n,k)\) and explore a duality with the \(sl_n\)-box-ball system. This paper is organized as follows: the first section is an introduction to the subject. In the second section, following [loc. cit.] the authors define the dynamical system and introduce \(\phi(n,k)\) (a family of dynamical system) and it’s tropicalization \(\Phi(n,k)\). They give the notion of soliton for \(\Phi(n,k)\), and state the first main result. In the third section, the explicit formula for \(\phi(n,k)\) and the relation to the geometric \(R\)-matrix is shown. In Section 4, by making use of the pictorial representation of reduced words in \(\widehat{S}_n\), the tau-function and the bilinear equation for the model are obtained. Sections 5 and 6 are devoted to computing soliton solutions for \(\Phi(n,k)\) in different two ways. The authors compute the tropicalization of the geometric solutions for \(\phi(n,k)\) obtained in [loc. cit.] in Section 5, and see that almost all geometric solutions vanish in tropicalization except for the simplest ones. In Section 6, they solve the tropical bilinear equations and prove the main result of the paper. In Section 7, they study the duality between the soliton solutions for \(\Phi(n,k)\) and the \(sl_n\)-box-ball system. After a brief introduction of the box-ball system, we present a conjecture which is a theorem for \(n=3,4\). They explain a strategy to prove it in subsection 7.3, and give the proof in the cases of \(n=3\) and \(4\) in subsection 7.4 and subsection 7.5 respectively. In the last section, the authors present other interesting numerical phenomena for \(\Phi(n,k)\), including negative solitons, pulsars, and relaxations of solitons and pulsars. The paper is supported by an appendix to explain the basics of the tropical semifield used in this paper.

MSC:

14H70 Relationships between algebraic curves and integrable systems
14T15 Combinatorial aspects of tropical varieties
68Q45 Formal languages and automata

Citations:

Zbl 1516.39007
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References:

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