A local characterization of crystals for the quantum queer superalgebra

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.

31 pages, 34 figures

References in corpus (1)

• $\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$

Cited by in corpus (7)

• $\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$
• Bumping operators and insertion algorithms for queer supercrystals
• Highest weight crystals for Schur Q-functions
• Candidate for the crystal $B(-\infty )$ for the queer Lie superalgebra
• A shifted Berenstein-Kirillov group and the cactus group
• Crystals for shifted key polynomials
• Primed decomposition tableaux and extended queer crystals