Reflection groups and cones of sums of squares

Sebastian Debus; Cordian Riener

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arXiv · 2011.09997

Reflection groups and cones of sums of squares

Sebastian Debus·Cordian Riener

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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.

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Sebastian Debus, Cordian Riener. 2020-11-19. Reflection groups and cones of sums of squares. https://arxiv.org/abs/2011.09997

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