Marvelous slices of orthogonal matrices

TL;DR

This paper explores the geometric decomposition of special orthogonal matrices in dimensions 4 and 5, revealing intricate structures and providing methods to identify real witness sets, with implications for understanding matrix spaces.

Contribution

It introduces a detailed decomposition of SO(4) and extends the approach to SO(5), uncovering geometric patterns and their limitations in higher dimensions.

Findings

Decomposition of SO(4) into 14 irreducible surfaces.

Identification of a real witness set for SO(4).

Extension of the decomposition approach to SO(5), but not SO(6).

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 $SO(6)$.