Comparing products of Schur functions and quasisymmetric functions

Abstract

In this thesis a conjecture of Okounkov, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity of certain differences of products of Schur functions are proved. In the first part of the work a combinatorial method is developed that allows to prove weaker versions of those conjectures. In the second part a recent result of Rhoades and Skandera is used to provide a proof of actual Schur positivity results. Several further generalizations are stated and proved. In particular, an intriguing log-concavity property of Schur functions is observed. In addition, a stronger conjecture is stated in language of alcoved polytops. A weaker version of this conjecture is proved using a characterization of Klyachko cone and the theory of Temperley-Lieb immanants.