Affine structures and a tableau model for E_6 crystals

TL;DR

This paper constructs a unique affine crystal structure for certain E_6^{(1)} Kirillov-Reshetikhin crystals using a generalized tableaux model and automorphisms, and proposes a conjecture for E_7^{(1)} crystals.

Contribution

It introduces a new tableaux model for classical E-type crystals and establishes the affine structure for specific E_6^{(1)} crystals, also conjecturing for E_7^{(1)}.

Findings

Unique affine crystal structure for E_6^{(1)} Kirillov-Reshetikhin crystals established.

Generalized tableaux model for classical E-type crystals introduced.

Conjecture proposed for E_7^{(1)} crystal structure.

Abstract

We provide the unique affine crystal structure for type E_6^{(1)} Kirillov-Reshetikhin crystals corresponding to the multiples of fundamental weights s Lambda_1, s Lambda_2, and s Lambda_6 for all s \geq 1 (in Bourbaki's labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type E and use the order three automorphism of the affine E_6^{(1)} Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type E_7^{(1)} Kirillov-Reshetikhin crystals corresponding to the adjoint node.