A characterization of matrix groups that act transitively on the cone of positive definite matrices (Q1322866)

It is well known that the group of all nonsingular lower block-triangular \(p\times p\) matrices acts transitively on the cone \({\mathcal P}^*\) of all positive definite \(p \times p\) matrices. In this paper the converse is established: If a matrix group acts transitively on \({\mathcal P}^*\), then its group algebra must be (similar to) the algebra of all lower block- triangular \(p \times p\) matrices with respect to a fixed partitioning.