Séminaire Lotharingien de Combinatoire, 86B.69 (2022), 12 pp.

Eric Marberg and Kam Hung Tong

Highest Weight Crystals for Schur $Q$-Functions

Abstract. Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra qn. Such qn-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur P-polynomials. We introduce a modified form of this category, with an extra crystal operator and a different tensor product, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur Q-polynomials. The crystals in this category have some interesting features not present for ordinary qn-crystals. For example, there is an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of normal qn-crystal structures on shifted tableaux and factorized reduced words. We describe extended forms of these structures that give examples in our new category.

Received: November 25, 2021. Accepted: March 4, 2022. Final version: April 1, 2022.

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