Permutation Representations and Automorphisms of Evolution Algebras - Costoya, Cristina; Mayorga-Pedraza, Pedro; Viruel-Arbaizar, Antonio Ángel (Springer, 2025)
We prove that the natural permutation representation of highly transitive finite groups cannot be realized as the full automorphism group of an idempotent, finite-dimensional evolution algebra acting on the set of lines spanned by its natural elements. Specifically, for any suf- ficiently large integer n and k ≥ 4, there does not exist an idempotent evolution algebra X of dimension n such that Aut(X) is isomorphic to a proper k-transitive subgroup of Sn. Nevertheless, we show that for any finite group G, any permutation representation ξ : G → Sn, and any field k, there exists an idempotent, finite-dimensional evolution k-algebra X such that Aut(X) ∼= G, and the induced representation of Aut(X) on the natural idempotents of X is equivalent to ξ.
Keywords: Grupos de permutaciones; Automorfismos; Grafos, Teoría de; Evolution algebra; Automorphism; Permutation group; Graph