slug: kaygorodov-martin-gonzalez-viruel-2026-evolution-automorphism-maximum
bibkey: kaygorodov2026evolution
doi: 10.48550/arXiv.2610.22248
url: https://arxiv.org/abs/2610.22248v1
triage: theorem
motivation_gids:
  - D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismControl
  - D5/S3/Combinatorics/Partitions/EvolutionAutomorphismWeight

The largest automorphism groups of perfect evolution algebras

Problem

Kaygorodov, Martín González and Viruel, Evolution algebras: a survey of results and a list of open problems, arXiv:2610.22248v1, section 2.3, Conjecture 3, pages 27–28, state:

Over C, the maximal order of the automorphism group of a perfect n-dimensional evolution algebra is max { product_l (l(2^l−1))^(m_l) m_l! : sum_l l m_l=n }, that is 1, 6, 21, 72, 155, 1296 for n=1,…,6 and n! for n>=7.

The dimensions are positive. Multiplication is bilinear and need not be associative. A natural basis has zero mixed products. Perfectness means that the span of all pair products is the whole algebra. Automorphisms are all multiplication-preserving complex linear equivalences.

Motivation

The source isolates the unresolved interaction between the diagonal kernel and the permutation image. Bounding these two parts separately by their independent maxima loses their shared support structure. EvolutionAutomorphismControl and EvolutionBasisTransport supply joint control for the full source family; EvolutionAutomorphismWeight supplies the exact positive-partition maximum and its shifted factorial absorption.

Gap

The source asserts attainment by direct sums of coefficient-one cyclic algebras and reports the upper bound checked by enumeration only through dimension four. The general joint kernel/image bound and the sharp universal inequality are separate from that reported computation. The source is the origin of the conjecture and of the attaining-family observation; the universal proof does not treat a conjectural assertion as a supporting theorem.

Route

Let P be the least successor-compatible quotient of the support relation, let beta be its successor permutation, and let k(B) be the size of an actual original fiber. Every k(B) is positive, their sum is n, and 1<=s=card(P)<=n. The actual diagonal kernel has the independent cycle-root cardinality. The permutation image is controlled jointly by the full centralizer of beta and the internal permutations of all fibers.

The root factor times the centralizer cardinality is exactly the weight of beta’s full partition of s, including singleton multiplicities. Ascending-factorial monotonicity gives product_B k(B)! <= (n−s+1)!, and shifted absorption gives maximumWeight(s)(n−s+1)! <= maximumWeight(n). Conjugacy along an arbitrary finite natural basis transfers the resulting full-group bound to every complex module of finrank n.

For attainment, the product P_sigma(x*y) on Fin(n) to C has squares e_i squared=e_(sigma(i)). Its invertible structure map makes it perfect. Its actual permutation image is exactly the centralizer of sigma: preservation implies commutation, and every commuting permutation has an explicit multiplication-preserving linear lift. The actual kernel is exactly the squaring assignments along sigma. The exact full-group order is therefore the full partition weight.

Only the maximizing cycle types are needed: identity in dimension one and in every dimension at least seven; single cycles of lengths two, three and five; two 2-cycles in dimension four; three 2-cycles in dimension six. The existing arithmetic identities supply all six small values and the factorial tail.

Falsifier

A perfect positive-dimensional complex evolution algebra whose full finite automorphism order exceeds the stated maximum would contradict the universal bound. Failure of an attaining witness to have the stated dimension, natural basis, perfection, or exact full-group order would invalidate attainment. A coordinate subgroup alone would not establish the assertion about the full group.

Evidence

The mathematical endpoint is D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismMaximum.result. Its first conjunct quantifies over arbitrary complex carriers, arbitrary finite basis index types, finrank equality, the natural-basis law and pair-product perfection. Its second conjunct constructs an actual n-dimensional commutative perfect evolution product with the same full-group cardinality bound as an equality.

The exact witness-group theorem is D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionPermutationFamily.permutation_aut_card; the universal inequality is D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionSharpBound.finite_and_bound. Lean kernel verification, scoped axiom closure and source binding determine their formal status. Publication and literature priority are separate from those mathematical statements.

Triage

What the settlement shows

Proved by D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismMaximum.result: the shared quotient size connects the diagonal root count to the permutation image. The positive fiber sizes account for all original vertices, and their shifted factorial budget absorbs the internal block-action kernel. This proves the source’s all-dimensional maximum without assuming a faithful quotient action or a graph lift. The source’s cyclic attainment observation is realized with actual products and full groups by D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionPermutationFamily.permutation_aut_card. Open here: Conjecture 4, which restricts to simple algebras, and the other conjectures in the survey. No claim about them follows from this settlement.

ASSUMED-UNVERIFIED

The source’s conjectural status refers to arXiv:2610.22248v1. Priority over later literature is not established here. The formal theorem does not assert originality of the cyclic attaining examples or a classification of every algebra attaining the maximum.