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Marvelous slices of orthogonal matrices

Taylor Brysiewicz, Fulvio Gesmundo

TL;DR

The paper reveals a remarkably organized decomposition of hollow special orthogonal varieties ${ m HSO}(4)$ and ${ m SO}^ullet(5)$ into a finite set of irreducible components whose incidence mirrors the face lattices of a cuboctahedron and a new polytope ${ m P}$, respectively. This structure yields concrete, fully real witness sets for ${ m SO}(4)$ and motivates a path toward similar real-algebraic descriptions for ${ m SO}(5)$, while showing that the same clean pattern fails for ${ m SO}(6)$. The authors relate component intersections to geometric features (edges and vertices) of the associated polytopes, and provide explicit parametrizations, circle and permutation-matrix intersections, and a thorough incidence analysis. Although a totally real witness set is achieved for ${ m SO}(4)$ and is conjectured in spirit for ${ m SO}(5)$, the work also highlights intrinsic obstructions in higher dimensions and introduces a novel polytope ${ m P}$ with rich combinatorial properties. Overall, the paper blends explicit algebraic geometry, combinatorial topology, and computational tools to illuminate deep regularities in the geometry of hollow orthogonal matrices and their real-slice structures.

Abstract

The space of $4 \times 4$ special orthogonal matrices with zeros on the diagonal decomposes into the union of $14$ irreducible surfaces whose intersections are beautifully encoded by the cuboctahedron. Using this decomposition, we exhibit a totally real witness set for $SO(4)$. We explain how to obtain a similar decomposition for $SO(5)$, where the $64$ components can be grouped to obtain such a correspondence with the face lattice of a $3$-polytope. We show that no such pattern exists for $SO(6)$.

Marvelous slices of orthogonal matrices

TL;DR

The paper reveals a remarkably organized decomposition of hollow special orthogonal varieties and into a finite set of irreducible components whose incidence mirrors the face lattices of a cuboctahedron and a new polytope , respectively. This structure yields concrete, fully real witness sets for and motivates a path toward similar real-algebraic descriptions for , while showing that the same clean pattern fails for . The authors relate component intersections to geometric features (edges and vertices) of the associated polytopes, and provide explicit parametrizations, circle and permutation-matrix intersections, and a thorough incidence analysis. Although a totally real witness set is achieved for and is conjectured in spirit for , the work also highlights intrinsic obstructions in higher dimensions and introduces a novel polytope with rich combinatorial properties. Overall, the paper blends explicit algebraic geometry, combinatorial topology, and computational tools to illuminate deep regularities in the geometry of hollow orthogonal matrices and their real-slice structures.

Abstract

The space of special orthogonal matrices with zeros on the diagonal decomposes into the union of irreducible surfaces whose intersections are beautifully encoded by the cuboctahedron. Using this decomposition, we exhibit a totally real witness set for . We explain how to obtain a similar decomposition for , where the components can be grouped to obtain such a correspondence with the face lattice of a -polytope. We show that no such pattern exists for .
Paper Structure (20 sections, 5 theorems, 24 equations, 7 figures, 1 table)

This paper contains 20 sections, 5 theorems, 24 equations, 7 figures, 1 table.

Table of Contents

  1. Main Result
  2. Related work and motivation
  3. The fourteen surfaces of hollow $\textrm{SO}(4)$
  4. Eight Spheres
  5. Six Tori
  6. Twenty-four curves and twelve pairs of points
  7. Twenty-four circles
  8. Twelve pairs of fixed point free signed permutation matrices
  9. A totally real witness set
  10. Extending to ${\textrm{SO}}(5)$
  11. The thirty-two tori and their intersections
  12. The thirty-two sections of ${\textrm{SO}}(3)$ and their intersections
  13. Two hundred fifty-six circles
  14. Two hundred eighty-eight signed permutation matrices
  15. A totally real witness set of $\textrm{SO}(5)$
  16. ...and 5 more sections

Key Result

Theorem 1.1

The variety $\textrm{HSO}(4) \subseteq \mathbb{R}^{16}$ is the union of $14$ irreducible surfaces: six tori of degree four and eight spheres of degree two. There exists a bijection between these components and the facets of the cuboctahedron $\mathcal{C}$ (illustrated on the left of fig:polytopes) s The relevant bijection is illustrated in fig:CuboctahedronFacets.

Figures (7)

  • Figure 1: The cuboctahedron $\mathcal{C}$ (left) and another polytope $\mathcal{P}$ (right). Their facets are in bijection with components of $\textrm{HSO}(4)$ and quadruples of components of $\textrm{SO}^\star(5)$ respectively. The face lattice of each gives the intersection lattice of the corresponding components in agreement with \ref{['thm:maintheorem']} and \ref{['thm:SO5']}.
  • Figure 2: The cuboctahedron with facets labeled by the diagrams identifying components of $\textrm{HSO}(4)$. This is the relevant bijection for \ref{['thm:maintheorem']}.
  • Figure 3: The vertices of $\mathcal{C}$ correspond to zero-dimensional intersections of components of $\textrm{HSO}(4)$. Each represents a signed double transposition and its negative.
  • Figure 4: The polytope $\mathcal{P}$ with facets labeled by diagrams identifying quadruples of components of ${\textrm{SO}}^\star(5)$ with the same zero pattern, giving the relevant bijection for \ref{['thm:SO5']}.
  • Figure 5: The vertices of $\mathcal{P}$ correspond to the zero-dimensional intersections of components of ${\textrm{SO}}^\star(5)$. Each represents $2^4$ signed permutation matrices.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 4.1
  • Theorem 5.1
  • Conjecture 5.2
  • Proposition 6.1
  • proof
  • Theorem 6.2
  • proof