Circular Free Spectrahedra

TL;DR

This paper classifies circular free spectrahedra, showing their defining LMIs have specific block structures, and characterizes noncommutative polynomials invariant under coordinate-wise unitary conjugation.

Contribution

It provides a complete classification of circular free spectrahedra and noncommutative polynomials invariant under certain unitary transformations.

Findings

Circular free spectrahedra have minimal LMI coefficients with a specific block structure.

Free circular matrix convex sets have minimal LMI coefficients with only two blocks.

Invariant noncommutative polynomials are direct sums of univariate polynomials.

Abstract

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by exp(i t): up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn-Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials.