Characterization of queer supercrystals

Abstract

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations of the type $A$ components of the queer crystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph $G$ on the type $A$ components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements.

39 pages, 8 figures

References in corpus (6)

• Highest weight modules over quantum queer Lie superalgebra U_q(q(n))
• Crystal Bases for the Quantum Superalgebra $U_q(gl(m,n))$
• Super duality and Crystal bases for quantum orthosymplectic superalgebras
• Super duality and Crystal bases for quantum orthosymplectic superalgebras II
• $\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$
• Crystal graphs for shifted tableaux

Cited by in corpus (7)

• Bumping operators and insertion algorithms for queer supercrystals
• Inequality of a class of near-ribbon skew Schur Q-functions
• Candidate for the crystal $B(-\infty )$ for the queer Lie superalgebra
• Crystals for shifted key polynomials
• Crystals for set-valued decomposition tableaux
• Primed decomposition tableaux and extended queer crystals
• A shifted Berenstein-Kirillov group and the cactus group