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Genetic algebras associated with sex linkage

Most work in genetic algebras has been concerned with inheritance which is symmetric with respect to sex, in that the characters studied are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. When asymmetric situations are studied, the development of the theory is complicated by the higher dimensions of the algebras, and by a feature to which Etherington (3, p. 40) drew attention, namely the fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. Etherington gave some results for the gametic and zygotic algebras of a single sex linked diallelic locus, and its properties were discussed further by Gonshor (4, p. 44). In a second paper (5, p. 334) Gonshor studied sex linkage in the case of multiple alleles, choosing a canonical basis which exhibited very clearly the multiplication table and ideal structure of the algebra. His treatment from the statement of the multiplication table in terms of the natural basis to its expression in terms of a canonical basis, is repeated in the displayed relations (4)–(8) below, for completeness and to establish the present notation.