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The authors formulate the Young-E(3) (YE3T) tensor product decomposition, coupling Schur-Weyl decomposition over Young subgroup-stabilized carriers with $SO(3)$ irreducible representations to directly construct complete, orthonormal rotation- and permutation-equivariant bases. Unlike standard Atomic Cluster Expansions (ACE) that are restricted to fully symmetric permutation carriers and often rely on overcomplete intermediates, YE3T directly spans arbitrary permutation characters and tensor product ranks. When applied to machine-learned interatomic potentials, this unified basis sets a new Pareto front, outperforming existing rotation-adapted architectures in both evaluation speed and predictive accuracy across atomistic and electronic-structure simulations.
Circumventing the overcomplete basis bottleneck of standard atomic cluster expansions, direct Schur-Weyl coupling of $S_N \times SO(3)$ representations establishes a new speed-accuracy Pareto front across atomistic and electronic-structure modeling.
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.