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<p>We investigate equivariant analogues of the Minkowski–Weyl theorem and Gordan’s lemma in an infinite-dimensional setting, where cones and monoids are invariant under the action of the infinite symmetric group. Building upon the framework developed in Kahle et al. (SIAM J Discrete Math 36(2):975–999, 2022) and Le-Römer (SIAM J Appl Algebra Geom 7(1):291–310, 2023), we extend the theory beyond the nonnegative case. Our main contributions include a local equivariant Minkowski–Weyl theorem, local–global principles for equivariant finite generation and stabilization of symmetric cones, and a full proof of the equivariant Gordan’s lemma conjectured in Le-Römer (SIAM J Appl Algebra Geom 7(1):291–310, 2023). We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.</p>

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On the Minkowski–Weyl theorem and Gordan’s lemma up to symmetry

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • Dinh Van Le

摘要

We investigate equivariant analogues of the Minkowski–Weyl theorem and Gordan’s lemma in an infinite-dimensional setting, where cones and monoids are invariant under the action of the infinite symmetric group. Building upon the framework developed in Kahle et al. (SIAM J Discrete Math 36(2):975–999, 2022) and Le-Römer (SIAM J Appl Algebra Geom 7(1):291–310, 2023), we extend the theory beyond the nonnegative case. Our main contributions include a local equivariant Minkowski–Weyl theorem, local–global principles for equivariant finite generation and stabilization of symmetric cones, and a full proof of the equivariant Gordan’s lemma conjectured in Le-Römer (SIAM J Appl Algebra Geom 7(1):291–310, 2023). We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.