Algebraically invisible selection: fecundity–viability regimes acting on evolution algebras

Abstract

An evolution algebra encodes uniparental inheritance: its natural basis indexes genotypes and its structure constants \(a_{ij}\) record the rate at which a parent of type \(i\) produces offspring of type \(j\). The classical genetic-algebra literature excludes selection from its scope; this paper appears to give the first treatment of selection inside the algebra. A selection regime is a pair \((u,v)\) of positive weights acting on parental fecundity and offspring viability, deforming the algebra into the principal isotope with structure matrix \(\operatorname{diag}(u)M\operatorname{diag}(v)\). We prove that, modulo regimes acting trivially, the selected algebra is isomorphic to the original precisely when \(u_iv_i^2\) is constant across genotypes, the exponent reflecting the quadratic character of uniparental reproduction. We compute the moduli space on a fixed digraph, identify the cycle monomials of the bipartite support graph — cross ratios \(a_{ij}a_{kl}/(a_{il}a_{kj})\) in the generic case — as the exact invariants of arbitrary selection, and obtain a doubly stochastic normal form through an essentially unique balancing regime. Non-degenerate selection preserves all graph-determined structure but not non-basic ideals; lethal selection creates evolutionarily closed genotype classes. A three-genotype drug-resistance pathway illustrates the results, including an invisible regime that reverses the population’s long-run composition.