Cartesian isotropic tensors revisited
Abstract
An alternative, streamlined methodology is presented for deriving the isotropic Cartesian tensor bases under the special orthogonal group SO(3). By shifting the traditional interpretation of the isotropy condition to analyze it as an algebraic system of equations for the rotation matrices themselves rather than the tensor components, lower-rank bases can be explicitly established in just a few lines without requiring finite coordinate rotations or complicated contractions of infinitesimal generators. Furthermore, a transparent combinatorial interpretation is provided for the number of tensors in the general higher-rank spanning sets. Finally, the Gram-matrix method is advocated as an efficient computational sieve to resolve the subsequent linear dependence.
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BibTeX
@misc{bouzas2026cartesian,
title = {{Cartesian isotropic tensors revisited}},
author = {Antonio O. Bouzas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04576},
eprint = {2609.04576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04576},
}
RIS
TY - UNPB
TI - Cartesian isotropic tensors revisited
AU - Antonio O. Bouzas
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04576
UR - https://arxiv.org/abs/2609.04576
AB - An alternative, streamlined methodology is presented for deriving the isotropic Cartesian tensor bases under the special orthogonal group SO(3). By shifting the traditional interpretation of the isotropy condition to analyze it as an algebraic system of equations for the rotation matrices themselves rather than the tensor components, lower-rank bases can be explicitly established in just a few lines without requiring finite coordinate rotations or complicated contractions of infinitesimal generators. Furthermore, a transparent combinatorial interpretation is provided for the number of tensors in the general higher-rank spanning sets. Finally, the Gram-matrix method is advocated as an efficient computational sieve to resolve the subsequent linear dependence.
ER -