Quantum nilpotent subalgebras of classical quantum groups and affine crystals

TL;DR

This paper explores the structure of quantum nilpotent subalgebras in type D quantum groups, revealing their affine crystal structure, providing a new polytope realization, and generalizing classical combinatorial correspondences.

Contribution

It demonstrates that the crystal of a quantum nilpotent subalgebra has an affine crystal structure and introduces a new polytope realization, extending combinatorial correspondences to type D.

Findings

Affine crystal structure of the subalgebra isomorphic to a limit of Kirillov-Reshetikhin crystals.

New polytope realization of $B^{n,s}$.

Generalization of Greene's formula for type D.

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 $s\geq 1$, 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 $D$.