An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_{2}^{(1)}$

TL;DR

This paper presents an explicit algorithm for a bijection between rigged configurations and highest weight paths in the $G_{2}^{(1)}$ adjoint crystals, facilitating combinatorial analysis of these structures.

Contribution

It introduces a novel explicit algorithm for the rigged configuration bijection specific to the $G_{2}^{(1)}$ adjoint crystal case, which was previously not explicitly constructed.

Findings

The algorithm preserves the static properties of rigged configurations.

It enables efficient computation of highest weight paths.

The method clarifies the combinatorial structure of $G_{2}^{(1)}$ crystals.

Abstract

We construct an explicit algorithm of the static-preserving bijection between the rigged configurations and the highest weight paths of the form $(B^{2,1})^{\otimes L}$ in the $G_{2}^{(1)}$ adjoint crystals.