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

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.