Highly symmetric matroids, the strong Rayleigh property, and sums of squares

TL;DR

This paper explores the strong Rayleigh property in symmetric matroids, providing criteria based on polynomial roots and sum of squares conditions, with implications for understanding matroid stability and structure.

Contribution

It offers new characterizations of strongly Rayleigh matroids with symmetry, linking polynomial root properties and sum of squares conditions to matroid stability.

Findings

Matroids with two orbits are strongly Rayleigh iff an associated polynomial has real roots.

Quadratic polynomial case yields explicit structural criteria.

Matroids with an orbit of rank two are strongly Rayleigh iff certain differences are sums of squares.

Abstract

We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szeg\H{o} theorem can be used to simplify the problem. When the Young subgroup has only two orbits, such a matroid is strongly Rayleigh if and only if an associated univariate polynomial has only real roots. When this polynomial is quadratic we get an explicit structural criterion for the strong Rayleigh property. Finally, if one of the orbits has rank two then the matroid is strongly Rayleigh if and only if the Rayleigh difference of any two points on this line is in fact a sum of squares.