[Paper Review] Linear orbits of alternating forms on real vector spaces

Leonid Ryvkin|arXiv (Cornell University)|Sep 7, 2016

Advanced Differential Equations and Dynamical Systems5 references3 citations

TL;DR

This paper completes the classification of GL(ℝⁿ)-orbits on Λᵏ(ℝⁿ)∗ by resolving the previously unaddressed cases (n,k) = (7,4) and (8,5). Using duality via volume forms and stability analysis, it establishes that Λ⁴(ℝ⁷)∗ has 20 orbits (15 non-degenerate, 4 stable), and Λ⁵(ℝ⁸)∗ has 35 orbits (31 non-degenerate, 3 stable), filling a critical gap in multisymplectic and special geometry.

ABSTRACT

In this note we complete the calculation of the number of $GL(\mathbb R^n)$-orbits on $Λ^k(\mathbb R^n)^*$, by treating the cases $(n,k)= (7,4)$ and $(8,5)$ not covered in the literature. We also calculate the number of of non-degenerate and stable orbits, as they are of special interest to multisymplectic and special geometry.

Motivation & Objective

• To complete the classification of GL(ℝⁿ)-orbits on Λᵏ(ℝⁿ)∗ for all n and k, particularly resolving the open cases (n,k) = (7,4) and (8,5).
• To determine the number of non-degenerate and stable orbits in these cases, as they are central to multisymplectic and special geometric structures.
• To establish a complete and consistent orbit count across all (n,k) pairs, integrating results from prior literature and filling gaps in the classification.
• To provide a rigorous proof framework using duality, contraction maps, and stabilizer analysis to distinguish orbit types under the GL(V) action.

Proposed method

• Utilizes the Hodge star isomorphism induced by a volume form Ω to relate orbits in Λᵏ(V∗) to those in Λⁿ⁻ᵏ(V), leveraging Lemma 3.1 to preserve orbit structure and stability.
• Applies the isomorphism c: Λᵏ(V) → Λⁿ⁻ᵏ(V∗), ξ ↦ ιξΩ, to translate orbit classification from multivectors to forms, especially for odd k and n−k.
• Employs Lemma 3.2 to relate degenerate orbits in Λᵏ(V∗) to orbits in Λᵏ(W∗) for codimension-1 subspaces W, enabling recursive counting of non-degenerate orbits.
• Analyzes stabilizer subgroups of multivector representatives to determine when c(ξ) and c(−ξ) represent the same orbit, using the presence of negative-determinant elements in Stab(ξ).
• Uses known orbit classifications for (n,k) = (8,3) and (7,3) to deduce results for (n,k) = (8,5) and (7,4) via duality and symmetry considerations.
• Relies on explicit stabilizer data from prior work [2] to determine which orbits are stable and which are not, based on whether their stabilizers contain elements of negative determinant.

Experimental results

Research questions

• RQ1 How many GL(ℝⁿ)-orbits exist on Λᵏ(ℝⁿ)∗ for the cases (n,k) = (7,4) and (8,5), which were previously unresolved?
• RQ2 What is the number of non-degenerate and stable orbits in Λ⁴(ℝ⁷)∗ and Λ⁵(ℝ⁸)∗, and how do they relate to the geometry of multisymplectic structures?
• RQ3 Can the orbit classification for alternating k-forms be completed using duality and stabilizer analysis, particularly when k and n−k are both odd?
• RQ4 How does the presence of negative-determinant elements in the stabilizer of a multivector affect the orbit structure of its Hodge dual in the dual form space?

Key findings

• For (n,k) = (7,4), the space Λ⁴(ℝ⁷)∗ has exactly 20 GL(ℝ⁷)-orbits, with 15 being non-degenerate and 4 being stable.
• For (n,k) = (8,5), the space Λ⁵(ℝ⁸)∗ has exactly 35 GL(ℝ⁸)-orbits, with 31 being non-degenerate and 3 being stable.
• The orbit count for Λ⁵(ℝ⁸)∗ is derived via duality from the known classification of Λ³(ℝ⁸), where 35 orbits and 31 non-degenerate orbits are established.
• The 20 orbits in Λ⁴(ℝ⁷)∗ arise from 14 distinct multivector orbits in Λ³(ℝ⁷), with 8 orbits from degenerate multivectors and 12 from non-degenerate ones, due to orbit doubling under sign reversal.
• Stable orbits in Λ⁴(ℝ⁷)∗ correspond to those multivectors in Λ³(ℝ⁷) whose stabilizers do not contain elements of negative determinant, resulting in 4 stable forms.
• The number of non-degenerate orbits in Λ⁴(ℝ⁷)∗ is confirmed via Lemma 3.2 to be 15, by relating degenerate forms to orbits in Λ⁴(ℝ⁶)∗, which has 4 orbits, and subtracting from the total 20.