Quantum nilpotent subalgebras of classical quantum groups and affine crystals

Abstract

We study the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$. We show that it has an affine crystal structure of type $D_n^{(1)}$ isomorphic to a limit of perfect Kirillov-Reshetikhin crystal $B^{n,s}$ for $s\ge 1$, and give a new polytope realization of $B^{n,s}$. We show that an analogue of RSK correspondence for type $D$ due to Burge is an isomorphism of affine crystals and give a generalization of Greene's formula for type $D$.

v5: 37 pages, the proof of Lemma 5.14(2) (Lemma 5.15(3) in v5) revised, an algorithm added in Section 5.4, Example 5.14 added, typos corrected; v4: 34 pages, reference corrected, minor correction, Proposition 3.2 is revised with minor corrections, comments added on Remark 3.3 and Remark 4.12, typos corrected, to appear in Journal of Combinatorial Theory, Series A

References in corpus (6)

• Kirillov--Reshetikhin crystals for nonexceptional types
• Quantum Unipotent Subgroup and dual canonical basis
• Combinatorial descriptions of the crystal structure on certain PBW bases
• RSK correspondence and classically irreducible Kirillov-Reshetikhin crystals
• Mirković-Vilonen Polytopes and a quiver construction of crystal basis in type $A$
• PBW bases and marginally large tableaux in type D

Cited by in corpus (5)

• Lusztig data of Kashiwara-Nakashima tableaux in type D
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$
• A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
• Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras
• Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with a minuscule node $r$