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Algebra, Geometry & Combinatorics: An SOS counterexample to an inequality of symmetric polynomials

Event Type
Mar 31, 2022   3:00 pm  
Isabelle Shankar
Gidon Orelowitz
Sum of squares (SOS) relaxations are often used to certify nonnegativity of polynomials and are equivalent to solving a semidefinite program (SDP). The feasible region of the SDP for a given polynomial is the Gram Spectrahedron. For symmetric polynomials, there are reductions to the problem size that can be done using tools from representation theory.  This gives rise to a smaller, more manageable spectrahedron, the Symmetry Adapted Gram Spectrahedron.  With this machinery, we disprove a 2011 conjecture about the complete homogeneous symmetric polynomials.  Specifically, we find an SOS counterexample to the claim that H_lambda <=  H_mu if and only if mu dominates lambda in partition dominance order.
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