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MiS Preprint
86/2019

An SOS counterexample to an inequality of symmetric functions

Alexander Heaton and Isabelle Shankar

Abstract

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. 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 paper we provide 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 rational 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.

Received:
05.09.2019
Published:
06.09.2019
MSC Codes:
05E05, 14P99, 90C22
Keywords:
sums of squares, symmetric functions

Related publications

inJournal
2021 Repository Open Access
Alexander Heaton and Isabelle Shankar

An SOS counterexample to an inequality of symmetric functions

In: Journal of pure and applied algebra, 225 (2021) 8, p. 106656