Young wall construction of level-1 highest weight crystals over $U_q(D_4^{(3)})$ and $U_q(G_2^{(1)})$

TL;DR

This paper presents a Young wall construction method for level-1 highest weight crystals over quantum affine algebras U_q(G_2^{(1)}) and U_q(D_4^{(3)}), utilizing path realization and affine energy functions.

Contribution

It introduces a novel Young wall construction based on four shapes of colored blocks for these specific quantum affine algebras.

Findings

Constructed Young walls for level-1 highest weight crystals.

Defined four types of colored blocks for the construction.

Provided a combinatorial model for crystals over the specified algebras.

Abstract

With the help of path realization and affine energy function, we give a Young wall construction of level-1 highest weight crystals $B(\lambda)$ over $U_{q}(G_{2}^{(1)})$ and $U_{q}(D_{4}^{(3)})$. Our construction is based on four different shapes of colored blocks, $\mathbf O$-block, $\mathbf I$-block, $\mathbf L$-block and $\mathbf{LL}$-block, obtained by cutting the unit cube in three different ways.