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Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra

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Abstract

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.

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Correspondence to Sami Assaf.

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Work was supported in part by a Simons Foundation Collaboration Grant for Mathematicians (Award 524477, S.A.).

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Assaf, S., Oguz, E.K. Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra. Ann. Comb. (2020) doi:10.1007/s00026-019-00477-0

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Keywords

  • Schur P-polynomials
  • Shifted tableaux
  • Crystals
  • Queer crystals

Mathematics Subject Classification

  • Primary 05E05
  • Secondary 05E10
  • 20G42