Reflection groups and cones of sums of squares

TL;DR

This paper explores cones of sums of squares forms invariant under reflection groups, utilizing representation theory and higher Specht polynomials to provide efficient descriptions and connect to non-negative forms, including a new proof for a specific case.

Contribution

It introduces a uniform description of cones of sums of squares invariant under reflection groups using higher Specht polynomials, enhancing understanding of their structure.

Findings

New description of cones using higher Specht polynomials

Connection established between non-negative forms and sums of squares

Provided a new proof for Harris's result on non-negative ternary even symmetric octic forms

Abstract

We consider cones of real forms which are sums of squares forms and invariant by a (finite) reflection group. We show how the representation theory of these groups allows to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the $A_{n}$, $B_n$, and $D_n$ case where we use so called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, for example, to study the connection of these cones to non-negative forms. In particular, we give a new proof of a result by Harris who showed that every non-negative ternary even symmetric octic form is a sum of squares.