Closed energy formulas on tensor squares of perfect crystals of types $D_4^{(3)}$ and $G_2^{(1)}$

Abstract

For each integer $l\ge1$, we give closed formulas for the local energies on tensor squares of the level-$l$ perfect crystals of types $D_4^{(3)}$ and $G_2^{(1)}$, normalized to vanish at the all-zero coordinate tensors. The $D_4^{(3)}$ energy is the maximum of three defects, equivalently of nine linear forms in parameters of classical highest-weight elements. Eliminating these parameters yields the maximum of nine piecewise-linear functions of the twelve input coordinates. Tensor-compatible similarity identifies the $G_2^{(1)}$ energy at level $l$ with one third of the pullback of the $D_4^{(3)}$ energy at level $3l$, yielding the same coordinate formula. The formula has no explicit level dependence, respects natural coordinate-set inclusions between levels, and allows direct evaluation without the combinatorial $R$-matrix or crystal-graph traversal.