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

2024/07/29 by Harris, Mitchell Tong
#FOS: Mathematics #Optimization and Control (math.OC) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2407.20200

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.

Related

• Gram-like matrix preserving extensions and completions of noncommutative polynomials
• Macaulay representation of the prolongation matrix and the SOS conjecture
• Positive Semidefinite Univariate Matrix Polynomials
• Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices
• Gram Matrices and Stirling numbers of a class of Diagram Algebras