Permutation representations and automorphisms of evolution algebras
Cristina Costoya, Pedro Mayorga, Antonio Viruel
TL;DR
The paper addresses whether permutation representations, especially highly transitive ones, can be realized as automorphism actions of finite-dimensional idempotent evolution algebras. It shows a negative result: for sufficiently large n and k ≥ 4, Aut(X) cannot be a proper k-transitive subgroup of S_n when acting on lines generated by a natural basis, hence forcing Aut(X) ≅ S_n in these cases. In contrast, it proves a universal positive construction: for any finite group G and any permutation representation ξ, there exist idempotent evolution algebras X with Aut(X) ≅ G and with the induced action on the natural idempotents equivalent to ξ. Moreover, using graph-based methods (CGV/CLTV), it establishes that any permutation representation on the set of idempotent elements ˜B_X can be realized, yielding infinitely many non-isomorphic X for any ξ. These results clarify the limits and possibilities of realizing group actions within evolution algebras and provide concrete tools to realize arbitrary permutation representations via graph-derived 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 $ξ\colon G \to S_n$, 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 $ξ$.