On the cones of classical groups

Abstract

The cone of a classical group $G$ is an affine $G\times G$-variety. The aim of this note is to initiate its combinatorial study in the cases when $G$ is the complex orthogonal or symplectic group. The coordinate ring of the cone of $G$ is a finitely generated commutative graded algebra. First the $G\times G$-module structure of its homogeneous components is determined. This is used to compute the Hilbert series of this coordinate ring in the cases when $G$ is the orthogonal group $\mathrm{O}(3)$, $\mathrm{O}(4)$, the special orthogonal group $\mathrm{SO}(4)$, and when $G$ is the symplectic group $\mathrm{Sp}(4)$. It is concluded that the coordinate ring of the cone of $\mathrm{O}(3)$ is not Koszul, hence the vanishing ideal of this cone has no quadratic Gr\"obner basis (although it is minimally generated by quadratic elements).