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Characterization of queer supercrystals. (English) Zbl 1435.05219

Summary: We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by D. Grantcharov et al. [J. Eur. Math. Soc. (JEMS) 17, No. 7, 1593–1627 (2015; Zbl 1336.17014)]. 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 S. Assaf and E. K. Oğuz [Sémin. Lothar. Comb. 80B, 80B.26, 12 p. (2018; Zbl 1411.05272)], with further axioms and a new graph \(G\) characterizing the relations of the type \(A\) components of the queer supercrystal. We provide a counterexample to Assaf’s and Oğuz’ 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. This also yields a new combinatorial description of the Schur expansion of the Schur \(P\)-polynomials.

MSC:

05E10 Combinatorial aspects of representation theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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