Fusion rules from root systems I: case ${\rm A}_n$

Axial algebras are commutative algebras generated by idempotents; they generalise associative algebras by allowing the idempotents to have additional eigenvectors, controlled by fusion rules. If the fusion rules are $\mathbb{Z}/2$-graded, axial algebras afford representations of transposition groups. We consider axial representations of Weyl groups of simply-laced root systems, which are examples of regular $3$-transposition groups. We introduce coset axes, a special class of idempotents based on embeddings of transposition groups, and use them to study the propagation of fusion rules in axial algebras, for root system ${\rm A}_n$. This is related to the construction of lattice vertex operator algebras and we show it reflects on the fusion of modules for the Virasoro algebra when we specialise our construction.