Crystal base of the negative half of quantum orthosymplectic superalgebra

TL;DR

This work constructs a crystal base for the negative half ${\mathcal U}(\mathfrak g)^-$ of a quantum orthosymplectic superalgebra, realized as a limit of crystals from $q$-oscillator representations and organized through parabolic Verma modules. The authors develop a PBW-type crystal realization and establish a sequence of compatible crystal bases for modules $V(\lambda,\ell)$, $P(\lambda)$, $Q(\lambda)$, and $R(\lambda)$, ultimately assembling ${\mathcal U}(\mathfrak g)^-$ and its crystal ${\mathscr B}(\infty)$. A central achievement is a combinatorial embedding of $\mathscr B(V(\lambda,\ell))$ into $\mathscr B(P(\lambda))$, mediated by separation algorithms and crystal isomorphisms: in type $B$ and $C$ via a super-analogue of the RSK correspondence, and in type $D$ via a Burge-type correspondence. The paper also proves a Burge theorem for orthosymplectic type, clarifying the interplay between crystal bases, Lusztig data, and parabolic inductions, thereby extending the oscillator- and parabolic-Verma-approach to nontrivial superalgebra settings with $n>0$. These results enrich the crystal-theoretic toolbox for quantum superalgebras and connect to broader combinatorial frameworks such as spinor models and RSK/Burge correspondences.

Abstract

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding.