Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra

We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P -polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) t...

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Bibliographic Details
Published inAnnals of combinatorics Vol. 24; no. 1; pp. 3 - 46
Main Authors Assaf, Sami, Oguz, Ezgi Kantarci
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2020
Springer Nature B.V
Subjects
Axioms
Combinatorics
Crystal structure
Crystals
Graphs
Mathematics
Mathematics and Statistics
Operators
Polynomials
Tensors
20G42
polynomials
Schur
Shifted tableaux
Crystals
Primary 05E05
Secondary 05E10
Queer crystals
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Summary:We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P -polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) to construct a connected queer crystal on semistandard shifted tableaux of a given shape. Using the tensor rule for queer crystals, this provides a new proof that products of Schur P -polynomials are Schur P -positive. Finally, to facilitate applications of queer crystals in the context of Schur P -positivity, we give local axioms for queer regular graphs, generalizing Stembridge’s axioms, that partially characterize queer crystals.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-019-00477-0