The algebraic approach to Bernstein's problem on Mendelian models of inheritance

Abstract

Consider a single gene locus for which the associated character is not sex-linked, and suppose that the character is exhibited as one of alleles. Under the Mendelian model of inheritance, if genotype frequencies are constant, the distribution of zygotes is stable from the second generation on, regardless of the value of n. (This is the celebrated Hardy–Weinberg law—see Hardy (1908).) In 1924 Bernstein raised the question of whether the Mendelian model is, in fact, a necessary consequence of the assumption that the zygote distribution is stable from the second generation on; when n = 2 or 3 Bernstein gave an affirmative solution, using elementary but tedious manipulation of probabilities. Lyubich (1971) used methods of convex analysis to obtain a partial solution of Bernstein's problem for general n. Holgate (1975) introduced algebraic methods in order to present a clearer picture of the underlying structures, and to relate Bernstein's problem to the algebras studied by Etherington (1939), Schafer (1949) and others. The purpose of this paper is to expound recent work of the author and others, building on Holgate's and Lyubich's ideas. The subject-matter of the paper is contained in a forthcoming joint paper by the author and A. E. Stratton.