The commutative Moufang loop module theory of elementary abelian $3$-groups

We consider the representations of the elementary abelian $3$-group $C_3^n$ in the category $\mathbf {CML}_3$ of commutative Moufang loops of exponent $3$. We determine a ring, $\mathcal {R}(C_3^n)$, such that $\mathcal {R}(C_3^n)$-modules are equivalent to abelian groups in $\mathbf {CML}_3/C_3^n$. Our main result presents $\mathcal {R}(C_3^n)$ as a quotient of the polynomial ring $\mathsf {GF}(3)[X_1, \dots X_{\binom {n}{2}}]$. We obtain generators for the regular representation of $\mathcal {R}(C_3^n)$ as a subring of the matrix ring $M_{\binom {n}{2}+1}(\mathsf {GF}(3))$.