Schur positivity and cell transfer

Thomas Lam, Alexander Postnikov, Pavlo Pylyavskyy

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We give combinatorial proofs that certain families of differences of products of Schur functions are monomial-positive. We show in addition that such monomial-positivity is to be expected of a large class of generating functions with combinatorial definitions similar to Schur functions. These generating functions are defined on posets with labelled Hasse diagrams and include for example generating functions of Stanley's (P, ω)-partitions. Then we prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions. This text contains the material from [LP, LPP].

Original languageEnglish (US)
Title of host publicationFPSAC 2006 - Proceedings
Subtitle of host publication18th Annual International Conference on Formal Power Series and Algebraic Combinatorics
Pages168-179
Number of pages12
StatePublished - Dec 1 2006
Event18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, CA, United States
Duration: Jun 19 2006Jun 23 2006

Other

Other18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006
CountryUnited States
CitySan Diego, CA
Period6/19/066/23/06

Keywords

  • Immanants
  • Kazhdan-Lusztig polynomials
  • Minors
  • Schur functions
  • Schur log-concavity
  • Schur positivity
  • Temperley-Lieb algebra

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