On the cones of classical groups
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909768765931520 |
|---|---|
| author | Domokos, Mátyás |
| author_facet | Domokos, Mátyás |
| contents | 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 Gröbner basis (although it is minimally generated by quadratic elements). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cones of classical groups Domokos, Mátyás Representation Theory Commutative Algebra Rings and Algebras Primary 13A50, Secondary 13P10, 14L35, 20G05, 20G42 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 Gröbner basis (although it is minimally generated by quadratic elements). |
| title | On the cones of classical groups |
| topic | Representation Theory Commutative Algebra Rings and Algebras Primary 13A50, Secondary 13P10, 14L35, 20G05, 20G42 |
| url | https://arxiv.org/abs/2509.03223 |