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

Abstract

A generalization of fixed-lattice cluster expansions (CE) and atomic cluster expansions (ACE) is presented that expresses both rotation and permutation symmetries. By performing Schur--Weyl decomposition in Young subgroup-stabilized carriers, followed by joint coupling of rotation and permutation representations, arbitrary $S_N\times SO(3)$ tensor product carriers are generated. This allows us to resolve complete, orthonormal joint rotation and permutation-adapted bases for arbitrary tensor product ranks and arbitrary angular and radial tensor product content. We show that this very general Young--E(3) (YE3T) tensor product basis contains the ACE basis as a subset, corresponding to the special case where permutation symmetry character is restricted to the fully symmetric carrier. Rather than constructing an overcomplete rotation and permutation-invariant basis and reducing it \textit{a posteriori}, Barthelemy \textit{et al.} recently demonstrated scaling benefits by not constructing an overcomplete basis. YE3T directly produces a complete orthonormal basis without an overcomplete step and rigorously extends to permutation characters beyond permutation-symmetric features. The YE3T decomposition yields new joint irreducible rotation- and permutation-equivariant basis sets that surpass existing rotation-adapted expansions in both speed and accuracy, defining a new Pareto front for machine-learned interatomic potentials. We show that the tunability of the permutation symmetry character makes the basis useful for both atomistic and electronic-structure simulations.