tensor-hexagonal_6bar-rank-3-axial.txt

Summary generated Tue Jul 22 05:55:43 PM CEST 2025 by TensSym for
point-symmetry group hexagonal_6bar, axial tensor of rank 3.
See https://github.com/hp35/tenssym for details.
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Symmetry operations associated with 6 (hexagonal, inversion symmetry) are:

1. 6-FOLD ROTATION SYMMETRY AROUND Z-AXIS WITH INVERSION APPLIED ALONG Z AXIS
6z (6-fold rotation symmetry around z-axis with inversion applied along z axis),
described by R(6z) for which det(R(6z)) = -1 (being an improper rotation):
⎡      √3     ⎤
⎢1/2   ──   0 ⎥
⎢      2      ⎥
⎢             ⎥
⎢-√3          ⎥
⎢────  1/2  0 ⎥
⎢ 2           ⎥
⎢             ⎥
⎣ 0     0   -1⎦

Compiling equation system from the 1 symmetries of point-symmetry group 6
(hexagonal, inversion symmetry).
Solving system of 26 equations for nonzero elements.
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POINT SYMMETRY GROUP 6 (HEXAGONAL, INVERSION SYMMETRY)

For the axial (pseudo) tensor of rank 3 (for which there in a completely non-
symmetrical case would be a maximum of 27 elements) under constraint of point-
symmetry group 6 (hexagonal, inversion symmetry), there are 21 nonzero elements,
of which 9 are independent.

The 9 independent and nonzero tensor elements are as follows:
       1.  xxx = -xyy = -yxy = -yyx
       2.  xxy = -yyy = xyx = yxx
       3.  xxz = yyz
       4.  xyz = -yxz
       5.  xzx = yzy
       6.  xzy = -yzx
       7.  zxx = zyy
       8.  zxy = -zyx
       9.  zzz = independent
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