Triply Transitive Natural Permutation Images of Idempotent Evolution Algebras

Source abstract

Let X be an n-dimensional idempotent evolution algebra over a field k, and let ρX: Aut(X) → Sn be the natural permutation representation determined by a natural basis. Costoya, Mayorga, and Viruel proved that, for n ≥ 3, 2-transitivity of the image of ρX implies that ρX is faithful, whereas, for n ≥ 5, 4-transitivity forces the image to be the full symmetric group. We classify the proper 3-transitive case left open by these results.If ρX(Aut(X)) ≤ Sn is 3-transitive, ρX(Aut(X)) ≠ Sn, and n ≥ 5, then n = q + 1 for a prime power q = pm ≡ 1 (mod 3), the field k contains a nontrivial cube root of unity, and char(k) ≠ p. Moreover, after a monomial change of natural basis, the structure matrix of X is a cubic-character matrix Cθ = (cxy)x,y∈P1(Fq), cxx = 0, cxy = θ(det(vx,vy)) for x ≠ y, where vx = (x,1)T for x ∈ Fq, v∞ = (1,0)T, and θ: Fq× → k× is a nontrivial character of order 3. The key rigidity comes from a compatibility relation derived from the evolution-algebra automorphism equation. Its exponent 2 reflects the squaring of a natural basis element: if χ is the character induced by the monomial lifts on a point stabilizer and s interchanges two points, then χ(s-1ks) = χ(k)2 for k in their common stabilizer. In the projective action this combines with inversion to force χ3 = 1.Conversely, every such cubic-character matrix defines an idempotent evolution algebra precisely when char k≠ p; indeed CθCθ-1 = qI. If Xθ denotes the resulting algebra, then Aut(Xθ) ≅ PGL2(q) ⋊ Gal(Fq/Fp)θ, where the subscript denotes the subgroup of field automorphisms fixing θ. Finally, for every nontrivial character η: Fq× → k× of order 3, Xθ ≅ Xη if and only if θ and η lie in the same Frobenius orbit. Thus the 4-transitivity threshold of Costoya–Mayorga–Viruel cannot be lowered to 3 in general. For q = 4, the construction yields the unique five-dimensional isomorphism class with automorphism group A5, and dimension 5 is minimal.