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

Event Type: Seminar/Symposium
Sponsor: N/A
Location: 347AH
Date: Mar 31, 2022 3:00 pm
Speaker: Isabelle Shankar
Contact: Gidon Orelowitz
E-Mail: gidono2@illinois.edu
Views: 56; 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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