Demazure crystals of generalized Verma modules and a flagged RSK correspondence

TL;DR

This paper establishes a crystal isomorphism between two realizations of generalized Verma modules using the RSK correspondence, extending its known properties and connecting to Demazure crystals and plane partitions.

Contribution

It proves the RSK correspondence is a $ ext{gl}_ ext{infty}$-crystal isomorphism for generalized Verma modules and introduces a flagged RSK version via Demazure crystal graphs.

Findings

RSK correspondence is a $ ext{gl}_ ext{infty}$-crystal isomorphism.

A flagged RSK correspondence is derived from Demazure crystal graphs.

Connection established between Demazure crystals and plane partitions.

Abstract

We prove that the Robinson-Schensted-Knuth correspondence is a $\gl_{\infty}$-crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of $\gl_{\infty}$. %This extends the previously known result that %the RSK correspondence is an isomorphism of bicrystals or double %crystals. A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition.