POLYNOMIAL IDENTITIES FOR BERNSTEIN ALGEBRAS OF SIMPLE MENDELIAN INHERITANCE

The Bernstein algebra D(n)G(alpha) of simple Mendelian inheritance is the n-fold duplicate of the gametic algebra G(alpha) with basis A(1),...,A(alpha) and structure constants A(i)A(j) = 1/2A(i) + 1/2A(j). We simplify and generalize results of Bernad et al. and Peresi by showing that every identity for G(alpha) follows from Costa's shape identity C, and that every identity for the zygotic algebra DG(alpha) follows from the recombination identity R. We use computer algebra to determine the identities of degree <= 7 for the copular algebra D(2)G(alpha).