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.

中文翻译：

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面向量子Querer超代数的晶体的局部表征

我们在半标准平移平台上定义算子，并对规则图使用Stembridge的局部表征来证明它们定义了晶体结构。这提供了新的证明Schur P-多项式为Schur正数。我们定义了酷儿晶体算子（也称为奇数Kashiwara算子），以在给定形状的半标准平移平台上构造连接的酷儿晶体。将张量法则用于酷儿晶体，这提供了新的证据，证明Schur P-多项式的乘积是Schur P-正的。最后，为方便在Schur P正性情况下应用奇尔晶体，我们给出了奇尔正则图的局部公理，对Stembridge公理进行了概括，该公理部分地表征了奇尔晶体。