A Characterization of Matrix Groups That Act Transitively on the Cone of Positive Definite Matrices

It is well known that the group of all nonsingular lower block -triangular pxp matrices acts transitively on the cone p* of all positive definite pxp matrices. This result has been applied to obtain several major reSUlts in mUltivariate statistical distribution theory and decision theory. Here a converse is established: if a matrix group acts transitively on P*, then its group algebra must be (similar to) the algebra of all lower block-triangUlar pxp matrices with respect to a fixed partitioning. This implies the nonexistence of multivariate normal linear statistical models with unrestricted covariance structure that admit a transitive group action, other than those classical models invariant under a block-triangular group.