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 $2^{r-1}$. In this article, we show that there exists a homogeneous cone of rank $r$ one of whose basic relative invariants has degree $2^{r-1}$. 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.