A family of irretractable square-free solutions of the Yang-Baxter equation

David Bachiller, Ferran Cedó, Eric Jespers, Jan Okniński

Research output: Contribution to journalArticleResearchpeer-review

18 Citations (Scopus)


© 2017 Walter de Gruyter GmbH, Berlin/Boston 2017. A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova's Strong Conjecture [19, Strong Conjecture 2.28 (I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.
Original languageEnglish
Pages (from-to)1291-1306
JournalForum Mathematicum
Issue number6
Publication statusPublished - 1 Nov 2017


  • Yang-Baxter equation
  • brace
  • set-theoretic solution


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