Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra
Eric Evert, Benjamin Passer
TL;DR
This paper investigates how matrix convex sets arising from free spectrahedra interact with their polar duals and with projections, focusing on real versus complex coefficient fields. It constructs and analyzes free spectrahedra generated by universal anticommuting self-adjoint unitaries, proving that for $g\ge3$ these sets are not the minimal matrix convex sets over the ball (and their duals are not maximal), with explicit free extreme points appearing at higher levels. It furthermore shows that the real free polar dual is rarely the projection of a real free spectrahedron, highlighting a sharp contrast with the complex case, and demonstrates that free spectrahedrops closed under complex conjugation can lack free extreme points. The paper also presents a class of complex free spectrahedra whose duals are again free spectrahedra, offering structural insights into when duals preserve free-spectrahedrality. Collectively, these results clarify the influence of the coefficient field and provide new boundaries between minimal/maximal matrix convex sets and their projections in the real and complex settings.
Abstract
We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.