Shifted positivity for queer square root crystals

Prove the shifted analogue of the character theorem for normal square root crystals associated with the queer Lie superalgebra $q_n$, namely that their characters are positive in the relevant $K$-theoretic Schur $P$- or shifted Grothendieck basis.

Background

The paper notes that a theory of normal square root crystals associated with the queer Lie superalgebra qnq_n already exists. These objects are related to qnq_n-crystals in a manner analogous to the relationship between the paper’s square root crystals and glngl_n-crystals.

The paper’s main theorem establishes Grothendieck positivity for characters of finite normal square root crystals of type glngl_n. The unresolved conjectural extension replaces Grothendieck positivity by positivity in a shifted, KK-theoretic Schur PP-function basis for the queer setting.

References

The first two authors conjectured a ``shifted'' analogue of Theorem~\ref{ch-thm} for these objects, in which $G$-positivity is replaced by {$-0.2mm$-positivity}; see Conj.~4.36.

— Grothendieck positivity for normal square root crystals  (2501.16640 - Marberg et al., 28 Jan 2025) in Section 1, subsection “Future directions and outline”

In the shifted setting, the corresponding questions are whether the pairings in arise from analogous Boolean fibers, how such fibers interact with crystal structures, and whether their coordinates admit a decorated vertex-model realization.

— Crystals and sign-reversing involutions for set-valued skew tableaux and decorated states of the five-vertex model  (2609.11233 - Shimazaki, 10 Sep 2026) in Section Conclusion