On the cones of classical groups

Domokos, Mátyás (2025) On the cones of classical groups. SERDICA MATHEMATICAL JOURNAL, 51 (3-4). pp. 367-390. ISSN 1310-6600

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Abstract

The cone of a classical group G is an affine G × 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 × 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 O(3), O(4), the special orthogonal group SO(4), and when G is the symplectic group Sp(4). It is concluded that the coordinate ring of the cone of 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). © 2025, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics. All rights reserved.

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