Schur positivity and cell transfer

Thomas Lam; Alexander Postnikov; Pavlo Pylyavskyy

Schur positivity and cell transfer

Thomas Lam, Alexander Postnikov, Pavlo Pylyavskyy

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference 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 language	English (US)
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
Country/Territory	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

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title = "Schur positivity and cell transfer",

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].",

keywords = "Immanants, Kazhdan-Lusztig polynomials, Minors, Schur functions, Schur log-concavity, Schur positivity, Temperley-Lieb algebra",

author = "Thomas Lam and Alexander Postnikov and Pavlo Pylyavskyy",

year = "2006",

month = dec,

day = "1",

language = "English (US)",

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TY - GEN

T1 - Schur positivity and cell transfer

AU - Lam, Thomas

AU - Postnikov, Alexander

AU - Pylyavskyy, Pavlo

PY - 2006/12/1

Y1 - 2006/12/1

N2 - 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].

AB - 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].

KW - Immanants

KW - Kazhdan-Lusztig polynomials

KW - Minors

KW - Schur functions

KW - Schur log-concavity

KW - Schur positivity

KW - Temperley-Lieb algebra

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M3 - Conference contribution

AN - SCOPUS:84860603513

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EP - 179

BT - FPSAC 2006 - Proceedings

T2 - 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006

Y2 - 19 June 2006 through 23 June 2006

ER -

Schur positivity and cell transfer

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