Abstract
In the recent past, linearly independent isotropic tensors of rank up to 6, under the compact rotation groups SO(2), SO(3) and SO(4) have been studied in some detail. The present paper extends these studies to the case of linearly independent isotropic tensors under the non-compact rotation groups SO(1, 1), SO(1, 2), SO(1, 3) and SO(2, 2). This is done by using the direct method of explicitly constructing these tensors, proving their linear independence and counting their numbers. Interestingly, it is found that these numbers are identical with the corresponding numbers for the case of the compact groups SO(2), SO(3) and SO(4).
Highlights
Linear invariants and isotropic tensors of the rotation groups SO(2), SO(3) and SO(4), have recently attracted the attention of many researchers
THE CASE r = 6:Here we choose the indices as i, j, k, l, m, n; the possible candidates for linearly independent isotropic tensors, are
It follows that in order to get all the linearly independent isotropic tensors of rank 6, we need to consider only those which are of the form
Summary
Linear invariants and isotropic tensors of the rotation groups SO(2), SO(3) and SO(4), have recently attracted the attention of many researchers. In the case d = 2, r = 4, there are 6 linearly independent isotropic tensors forming a complete set, which may be chosen to be (2.1, 2.3). THE CASE r = 6:Here we choose the indices as i, j, k, l, m, n; the possible candidates for linearly independent isotropic tensors, are
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