Comparing products of Schur functions and quasisymmetric functions
Massachusetts Institute of Technology. Dept. of Mathematics.
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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.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 71-74).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology