Primitive set-theoretic solutions of the Yang–Baxter equation
Ferran Cedó, Eric Jespers, Jan Okniński
Abstract
To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set [Formula: see text] there is a naturally associated finite solvable permutation group [Formula: see text] acting on [Formula: see text]. We prove that every primitive permutation group of this type is of prime order [Formula: see text]. Moreover, [Formula: see text] is then a so-called permutation solution determined by a cycle of length [Formula: see text]. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.
Topics & Concepts
MathematicsPermutation (music)Permutation groupDegenerate energy levelsSet (abstract data type)CombinatoricsGroup (periodic table)Yang–Baxter equationPrime (order theory)Type (biology)Order (exchange)Perspective (graphical)Pure mathematicsDiscrete mathematicsGeometryChemistryComputer scienceFinanceEcologyQuantumEconomicsOrganic chemistryQuantum mechanicsPhysicsBiologyProgramming languageAcousticsAlgebraic structures and combinatorial modelsAdvanced Topics in AlgebraFinite Group Theory Research