The structure of Gram matrices of sum of squares polynomials with restricted harmonic support

TL;DR

This paper explores the structured Gram matrices of SOS polynomials with restricted harmonic support, connecting representation theory of SL(2) and SO(2) to reveal new structural insights.

Contribution

It establishes a novel link between SOS polynomial Gram matrices and representation theory, showing how invariant subspaces influence their structure.

Findings

SOS polynomials in invariant subspaces have structured Gram matrices

The structure relates SL(2) and SO(2) representation theories

Tools developed are of independent mathematical interest

Abstract

Some sum of squares (SOS) polynomials admit decomposition certificates, or positive semidefinite Gram matrices, with additional structure. In this work, we use the structure of Gram matrices to relate the representation theory of $SL(2)$ to $SO(2)$. Informally, we prove that if $p$ is a sum of squares and lives in some of the invariant subspaces of $SO(2)$, then it has a positive semidefinite Gram matrix that lives in certain invariant subspaces of $SL(2)$. The tools used in the proof construction are of independent interest.