Fusion rules for V_{L_2}^{S_4}

This paper determines the fusion rules for the fixed-point vertex operator algebra $V_{L_2}^{S_4}$, where $L_2$ denotes the rank-one even lattice satisfying $(α,α)=2$. The main idea is to identify certain irreducible $V_{\mathbb Zζ}^{+}$-modules as direct sums of irreducible $V_{L_2}^{S_4}$-modules. These identifications allow us to compute the relevant entries of the $S$-matrix for $V_{L_2}^{S_4}$. Applying the Verlinde formula, we then obtain the complete fusion rules for $V_{L_2}^{S_4}$.