Robinson-Schensted-Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

TL;DR

This paper introduces a novel construction of standard modules for general linear groups over non-archimedean fields using a modified Robinson-Schensted-Knuth correspondence, linking quantum group bases with matrix polynomial bases.

Contribution

It develops a new approach to representation theory by modifying the RSK correspondence to construct standard modules, connecting dual canonical bases with DRS bases.

Findings

Constructed new standard modules for GL over non-archimedean fields.

Established a link between dual canonical bases and DRS bases.

Provided a categorification of matrix polynomial ring bases.

Abstract

We construct new "standard modules" for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky's multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.