Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

TL;DR

This paper provides a combinatorial realization of crystal bases for modified quantized enveloping algebras of types A+∞ and A∞, leading to a generalized RSK correspondence through a bicrystal structure.

Contribution

It introduces a new combinatorial model using integral bimatrices and tableaux to realize crystal bases and their Peter-Weyl decomposition, extending classical RSK.

Findings

Realization of crystal bases as integral bimatrices.

Decomposition of tensor products into extremal weight crystals.

A generalized RSK correspondence via bicrystal structures.

Abstract

The crystal base of the modified quantized enveloping algebras of type $A_{+\infty}$ or $A_\infty$ is realized as a set of integral bimatrices. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspondence.