An example of homogeneous cones whose basic relative invariant has maximal degree

It is known that degrees of basic relative invariants of homogeneous open convex cones of rank r are less than or equal to 2ʳ⁻¹. In this article, we show that there exists a homogeneous cone of rank r one of whose basic relative invariants has degree 2ʳ⁻¹. The main idea for this is to construct such a homogeneous cone inductively to have specific structure constants which enable us to calculate degrees of its basic relative invariants. We study homogeneous cones of rank $3$ in detail in order to see non-triviality of the existence of homogeneous cones with given structure constants.