Permutation representations and automorphisms of evolution algebras

Abstract: 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 sufficiently large integer $n$ and $k \geq 4$, there does not exist an idempotent evolution algebra $X$ of dimension $n$ such that $\operatorname{Aut}(X)$ is isomorphic to a proper $k$-transitive subgroup of $S_n$. Nevertheless, we show that for any finite group $G$, any permutation representation $\xi \colon G \to S_n$, and any field $\Bbbk$, there exists an idempotent, finite-dimensional evolution $\Bbbk$-algebra $X$ such that $\operatorname{Aut}(X) \cong G$, and the induced representation of $\operatorname{Aut}(X)$ on the natural idempotents of $X$ is equivalent to $\xi$.