The Young-E(3) Tensor Product Decomposition for Rotation and Permutation Equivariant Cluster Expansions

This paper presents a generalization of fixed-lattice cluster expansions and atomic cluster expansions that jointly expresses rotation and permutation symmetries. It applies Schur-Weyl decomposition in Young subgroup-stabilized carriers, then couples rotation and permutation representations to generate arbitrary S_N × SO(3) tensor-product carriers. The resulting framework provides complete, orthonormal joint rotation- and permutation-equivariant representations for atomistic cluster-expansion modeling.