# Joint Invariants of Linear Symplectic Actions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Recollection: Invariants

#### 2.1. Smooth Invariants

#### 2.2. Polynomial Invariants

#### 2.3. Rational Invariants

#### 2.4. Our Setup

#### 2.5. The Equivalence Problem

## 3. Invariants on Symplectic Vector Spaces

#### 3.1. The Case $n=1$

#### 3.1.1. $V\times V$

#### 3.1.2. ${V}^{\times \phantom{\rule{3.33333pt}{0ex}}3}=V\times V\times V$

#### 3.1.3. ${V}^{\times \phantom{\rule{3.33333pt}{0ex}}4}$

#### 3.1.4. ${V}^{\times \phantom{\rule{3.33333pt}{0ex}}5}$

#### 3.1.5. General ${V}^{\times \phantom{\rule{3.33333pt}{0ex}}m}$

#### 3.2. The General Case: Algebra of Polynomial Invariants

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

#### 3.3. The General Case: Field of Rational Invariants

## 4. Variation on the Group and Space

#### 4.1. Symmetric Joint Invariants

**Theorem**

**2.**

**Proposition**

**1.**

**Proof.**

#### 4.2. Conformal and Affine Symplectic Groups

#### 4.3. Invariants in the Contact Space

#### 4.3.1. The Case $n=1$

#### 4.3.2. The General Case

## 5. From Joint to Differential Invariants

#### 5.1. Jets of Curves in Symplectic Vector Spaces

#### 5.2. Symplectic Discretization

#### 5.3. Contact Discretization

#### 5.4. Functions and Other Examples

## 6. Relation to Binary and Higher Order Forms

**Remark**

**2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hilbert, D. Theory of Algebraic Invariants; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Olver, P.J. Classical invariant theory. In London Mathematical Society Student Texts; Cambridge University Press: Cambridge, UK, 1999; Volume 44. [Google Scholar]
- Popov, V.L.; Vinberg, E.B. Invariant theory. In Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences; Springer: Berlin, Germany, 1994; Volume 55. [Google Scholar]
- Rosenlicht, M. Some basic theorems on algebraic groups. Am. J. Math.
**1956**, 78, 401–443. [Google Scholar] [CrossRef] - Weyl, H. Classical Groups; Princeton University Press: Princeton, NJ, USA, 1946. [Google Scholar]
- Kruglikov, B.; Lychagin, V. Global Lie-Tresse theorem. Selecta Math.
**2016**, 22, 1357–1411. [Google Scholar] [CrossRef] [Green Version] - Jensen, J.O.; Kruglikov, B. Differential Invariants of Linear Symplectic Actions. Symmetry
**2020**, 12, 2023. [Google Scholar] [CrossRef] - Bibikov, P.; Lychagin, V. GL
_{2}(ℂ)-orbits of Binary Rational Forms. Lobachevskii J. Math.**2011**, 32, 95–102. [Google Scholar] [CrossRef] - Sylvester, J.J. An Essay on Canonical Forms; George Bell: 1851; On a Remarkable Discovery in the Theory of Canonical Forms and of Hyperdeterminants; Philosophical Magazine: 1851; Mathematical Papers; Chelsea: New York, NY, USA, 1973; pp. 34, 203–216, 41, 265–283. [Google Scholar]
- Alexander, J.; Hirschowitz, A. Polynomial interpolation in several variables. J. Algebr. Geom.
**1995**, 4, 201–222. [Google Scholar] - Andreassen, F. Joint Invariants of of Symplectic and Contact Lie Algebra Actions. Master’s Thesis in Mathematics, UiT the Arctic University of Norway, Tromsø, Norway, June 2020. Available online: https://hdl.handle.net/10037/19003 (accessed on 20 August 2020).
- Mumford, D.; Fogarty, J.; Kirwan, F. Geometric Invariant Theory; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Eisenbud, D. The geometry of syzygies: A second course in algebraic geometry and commutative algebra. In Graduate Texts in Mathematics; Springer: Berlin/Heidelberg, Germany, 2006; Volume 229. [Google Scholar]
- Olver, P.J. Joint invariant signatures. Found. Comput. Math.
**2001**, 1, 3–68. [Google Scholar] [CrossRef] [Green Version] - Ishikawa, M.; Wakayama, M. Minor summation formula of Pfaffians, Survey and a new identity. Adv. Stud. Pure Math.
**2000**, 28, 133–142. [Google Scholar] - Vust, T. Sur la théorie des invariants des groupes classiques. Ann. Inst. Fourier
**1976**, 26, 1–31. [Google Scholar] [CrossRef] [Green Version] - Springer, T.A. Invariant Theory. In Lecture Notes in Math; Springer: Berlin/Heidelberg, Germany, 1977; Volume 585. [Google Scholar]
- Bibikov, P.; Lychagin, V. Classification of linear actions of algebraic groups on spaces of homogeneous forms. Dokl. Math.
**2012**, 85, 109–112. [Google Scholar] [CrossRef] - Gün Polat, G.; Olver, P.J. Joint differential invariants of binary and ternary forms. Port. Math.
**2019**, 76, 169–204. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Andreassen, F.; Kruglikov, B.
Joint Invariants of Linear Symplectic Actions. *Symmetry* **2020**, *12*, 2020.
https://doi.org/10.3390/sym12122020

**AMA Style**

Andreassen F, Kruglikov B.
Joint Invariants of Linear Symplectic Actions. *Symmetry*. 2020; 12(12):2020.
https://doi.org/10.3390/sym12122020

**Chicago/Turabian Style**

Andreassen, Fredrik, and Boris Kruglikov.
2020. "Joint Invariants of Linear Symplectic Actions" *Symmetry* 12, no. 12: 2020.
https://doi.org/10.3390/sym12122020