Combinatorial Theory

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra \(\mathfrak{q}_n\). Such \(\mathfrak{q}_n\)-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur \(P\)-polynomials. This article studies a modified form of this category, 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 \(\mathfrak{q}_n\)-crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of \(\mathfrak{q}_n\)-crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.

Mathematics Subject Classifications: 05E05, 05E10

Keywords: Crystals, Schur \(Q\)-functions, queer Lie superalgebras, shifted tableaux, involution words