Zero Action on Perfect Crystals for U_q(G_2^{(1)})

TL;DR

This paper explicitly describes the zero Kashiwara operators on certain perfect crystals for the quantum affine algebra U'_q(G_2^{(1)}), demonstrating their coherence and structure.

Contribution

It provides an explicit description of zero Kashiwara operators on U'_q(G_2^{(1)})-crystals using a similarity technique, establishing the coherence of the crystal family.

Findings

Zero Kashiwara operators are explicitly described.

The crystal family {B_l} is shown to be coherent.

The technique relates to a similar approach for U'_q(D_4^{(3)})-crystals.

Abstract

The actions of 0-Kashiwara operators on the $U'_q(G_2^{(1)})$-crystal $B_l$ in [Yamane S., J. Algebra 210 (1998), 440-486] are made explicit by using a similarity technique from that of a $U'_q(D_4^{(3)})$-crystal. It is shown that $\{B_l\}_{l\ge1}$ forms a coherent family of perfect crystals.