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Vol. 51 No. 3–4 (2025)
Serdica Mathematical Journal

Articles

On the cones of classical groups

https://doi.org/10.55630/serdica.2025.51.367-390

Published 23-12-2025

  • Mátyás Domokos

Mátyás Domokos
Alfréd Rényi Institute of Mathematics, Hungary

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 case of the general linear group being well documented in the literature). 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).

2020 Mathematics Subject Classification:

13A50, 13P10, 14L35, 20G05, 20G42

Keywords

orthogonal group, symplectic group, Hilbert series, ideal of relations, quantum groups, Gröbner basis

Full text

Author Details

Mátyás Domokos

HUN-REN Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13-15
1053 Budapest, Hungary
e-mail: domokos.matyas@renyi.hu

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