Diversification, a central issue in the study of capital allocation, has much to do with symmetries and asymmetries in the distribution of asset returns. A diversified portfolio imposes symmetry on the allocation vector in order to balance out much of the asymmetries in the returns vector. Using group and majorization theory, we explore what can be established about the allocation vector when the asymmetries in the returns vector are carefully controlled. The key insight is that preferences over allocations can be partially ordered via majorized convex hulls that have been generated by group elements. It is shown that transitive permutation groups, rather than the more structured permutation symmetric group, suffice to ensure complete portfolio diversification. Point-wise stabilizer subgroups admit separability in the allocation of funds across sectors. When, together with imperfect symmetry in the sources of randomness, asset returns differ by heterogeneity in location or scale parameters then we bound the admissible allocation vector by a set of linear constraints. For a distribution that is symmetric under reflection groups, the linear constraints may be further strengthened whenever there exists an hyperplane that separates convex sets.
