## 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 language | English (US) |
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Title of host publication | FPSAC 2006 - Proceedings |

Subtitle of host publication | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 168-179 |

Number of pages | 12 |

State | Published - Dec 1 2006 |

Event | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, CA, United States Duration: Jun 19 2006 → Jun 23 2006 |

### Other

Other | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 |
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Country | United States |

City | San Diego, CA |

Period | 6/19/06 → 6/23/06 |

## Keywords

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