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 SN× 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.
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BibTeX
@misc{goff2026younge3,
title = {{The Young-E(3) Tensor Product Decomposition for Rotation and Permutation Equivariant Cluster Expansions}},
author = {James M. Goff and Aidan P. Thompson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31895},
eprint = {2609.31895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31895},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Young-E(3) Tensor Product Decomposition for Rotation and Permutation Equivariant Cluster Expansions
AU - James M. Goff
AU - Aidan P. Thompson
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31895
UR - https://arxiv.org/abs/2609.31895
AB - 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× 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.
ER -