Recurrence relations for linear mating schemes.

*(English)*Zbl 0569.92013In this paper the author applies the methods of genetic algebras to study an assortative mating scheme. This is of interest since assorting is especially difficult to deal with by these methods and thus has not been dealt with much algebraically up to now.

The author considers the copular algebra corresponding to a diploid locus with two alleles. (The copular algebra is the duplicate of the zygotic algebra. This has rarely been studied but is a natural object to deal with if one is interested in assortative mating.) The product for the algebra corresponding to assortative mating is obtained from the product for the copular algebra by following the latter product by a linear map. (This is analogous to what has been done for the algebras with mutations.)

First it is shown that one special case does lead to a genetic algebra. Afterwards a more general situation is considered where one does not obtain a genetic algebra. Now D. McHale and G. R. Ringwood [J. Lond. Math. Soc., II. Ser. 28, 17-26 (1983; Zbl 0515.17010)] have shown that for a genetic algebra the plenary operator can always be linearized. (The plenary map is the operator which takes the genotype distribution for one generation into the distribution for the following generation. Although this is ordinarily a quadratic map there is a standard trick for making it linear by enlarging the appropriate vector space.) Although their method of linearizing does not work in this case, it is shown that linearization is possible if one uses a certain infinite dimensional vector space.

There appears to be a minor computational error in table (14). It seems that the coefficient of \(C_{22}\) in \(C^ 2_{11}\) should not be (1/16)\(\gamma\). However, minor errors such as this do not detract from the exciting fact that the author has demonstrated that assortative mating can be dealt with by the methods of genetic algebras. (This always seemed to be a topic which must be left to the classical methods of probability theory.)

The author considers the copular algebra corresponding to a diploid locus with two alleles. (The copular algebra is the duplicate of the zygotic algebra. This has rarely been studied but is a natural object to deal with if one is interested in assortative mating.) The product for the algebra corresponding to assortative mating is obtained from the product for the copular algebra by following the latter product by a linear map. (This is analogous to what has been done for the algebras with mutations.)

First it is shown that one special case does lead to a genetic algebra. Afterwards a more general situation is considered where one does not obtain a genetic algebra. Now D. McHale and G. R. Ringwood [J. Lond. Math. Soc., II. Ser. 28, 17-26 (1983; Zbl 0515.17010)] have shown that for a genetic algebra the plenary operator can always be linearized. (The plenary map is the operator which takes the genotype distribution for one generation into the distribution for the following generation. Although this is ordinarily a quadratic map there is a standard trick for making it linear by enlarging the appropriate vector space.) Although their method of linearizing does not work in this case, it is shown that linearization is possible if one uses a certain infinite dimensional vector space.

There appears to be a minor computational error in table (14). It seems that the coefficient of \(C_{22}\) in \(C^ 2_{11}\) should not be (1/16)\(\gamma\). However, minor errors such as this do not detract from the exciting fact that the author has demonstrated that assortative mating can be dealt with by the methods of genetic algebras. (This always seemed to be a topic which must be left to the classical methods of probability theory.)

Reviewer: H.Gonshor