Two-level Bernsteinian populations

In this article we give a partial solution to Bernstein's problem concerning quadratic mappings of a simplex into itself: the description of all those mappings $V$ satisfying the condition $V^2=V$. The formulation of this question is connected with mathematical genetics. In this article, the selected class of mappings is characterized by the existence of linear preservation laws sufficient for the parametrization of a mapping. In genetic terms, this means the existence of genes (but, generally speaking, with a more complex behavior than in classical genetics). This article describes all these mappings.
Bibliography: 8 titles.