An SOS Counterexample to an Inequality of Symmetric Functions

Alex Heaton
Max Planck Institute for Mathematics in Science
Postdoctoral Researcher

“It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. Special cases of these inequalities go back to Maclaurin, Newton, and Schur, for example. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this talk, we describe a counterexample, showing that homogeneous symmetric functions break the pattern. We use a semidefinite program to find a positive semidefinite matrix whose factorization provides an explicit sums of squares decomposition of the polynomial H44 − H521 as a sum of 41 squares. This certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares of other polynomials cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order. This is joint work with Isabelle Shankar of UC-Berkeley.”

Light refreshments will be served @ 1:30pm in EMS E424A.