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Minkowski-Weyl theorem and Gordan's lemma up to symmetry

Dinh Van Le

TL;DR

This work extends the classical Minkowski-Weyl duality and Gordan's lemma to the realm of Sym-invariant cones and monoids in the infinite-dimensional space R^(N). By developing a local equivariant Minkowski-Weyl theorem, establishing local-global principles for finite generation and stabilization, and proving a full equivariant Gordan's lemma, the paper provides a coherent framework for understanding symmetric convex and lattice structures up to Sym. It also classifies non-pointed cones and non-positive normal monoids, and outlines open problems regarding dual objects, thereby advancing invariant theory and polyhedral geometry in infinite dimensions. The results rely on stabilizing chains, dual-cone analysis, and constructive generation methods, with potential implications for algorithmic and computational aspects in symmetry-reduced convex and lattice problems.

Abstract

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 earlier, 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. We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.

Minkowski-Weyl theorem and Gordan's lemma up to symmetry

TL;DR

This work extends the classical Minkowski-Weyl duality and Gordan's lemma to the realm of Sym-invariant cones and monoids in the infinite-dimensional space R^(N). By developing a local equivariant Minkowski-Weyl theorem, establishing local-global principles for finite generation and stabilization, and proving a full equivariant Gordan's lemma, the paper provides a coherent framework for understanding symmetric convex and lattice structures up to Sym. It also classifies non-pointed cones and non-positive normal monoids, and outlines open problems regarding dual objects, thereby advancing invariant theory and polyhedral geometry in infinite dimensions. The results rely on stabilizing chains, dual-cone analysis, and constructive generation methods, with potential implications for algorithmic and computational aspects in symmetry-reduced convex and lattice problems.

Abstract

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 earlier, 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. We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.
Paper Structure (15 sections, 27 theorems, 126 equations)

This paper contains 15 sections, 27 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Symmetric cones and monoids
  3. Symmetric groups and their actions
  4. Symmetric cones
  5. Symmetric monoids
  6. Equivariant Minkowski--Weyl theorem
  7. Dual cones
  8. Equivariant Minkowski--Weyl theorem: global extension
  9. Equivariant Minkowski--Weyl theorem: local version
  10. Local-global principles
  11. Non-pointed symmetric cones and non-positive symmetric monoids
  12. Non-pointed symmetric cones
  13. Non-positive symmetric normal monoids
  14. Equivariant Gordan's lemma
  15. Further open problems

Key Result

Proposition 3.1

Let $C\subseteq\mathbb{R}^{(\mathbb{N})}$ be any $\mathop{\mathrm{Sym}}\nolimits$-invariant cone. Then the restricted dual cone $C^*\cap \mathbb{R}^{(\mathbb{N})}$ is one of the following: In particular, $C^*\cap \mathbb{R}^{(\mathbb{N})}$ is always $\mathop{\mathrm{Sym}}\nolimits$-equivariantly finitely generated.

Theorems & Definitions (64)

  • Conjecture 1.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['p:global-W-M']}
  • Theorem 3.3
  • Example 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 54 more