Commutation symmetries of tensor indices
Let us consider covariant tensors T of order r over a finitedimensional vector space V. A commutation symmetry of such tensors is a pair (C , e) where C is a subgroup of the symmetric group S_{r} and e is a homomorphism
of C onto a finite subgroup of the group S^{1} of complex units.
We say that a covariant tensor T of order r possesses a symmetry (C , e) if T fulfils

"c Î C , "v_{1} , ¼, v_{r} Î V : T(v_{c(1)} , ¼, v_{c(r)}) 


e(c) T(v_{1} , ¼, v_{r} ) . 

 

If a tensor T has a symmetry (C , e) and
T¢ is an isomer



T(v_{p(1)} , ¼, v_{p(r)}) 

 

of T defined by a fixed permutation p Î S_{r}, then T¢ possesses the symmetry
(C¢, e¢) which is given by



"c¢ Î C¢: e¢(c¢) = e(p^{1} °c¢°p) . 

 

Since T¢ differs from T only in a permutation of its arguments, the symmetry
(C¢, e¢) is no ''news'' in comparison with (C , e).
Thus it is natural to regard the symmetries (C , e) and (C¢, e¢) as equivalent.
I present here a
HTML version
of tables of all commutation symmetries belonging to subgroups of
S_{r} with r
£ 6. The tables are complete up to equivalence of symmetries. They were calculated by means of my Mathematica package
PERMS.
The text about the tables is a part of my postdoctoral thesis.
The HTML version of the tables is not legible under the Netscape Communicator 2.x since
certain fonts can not be used. If a later version of the Netscape Communicator does not yield a correct result, then
modify the entries of your files .Xdefaults or .Xresources by means of the
instructions
of Ian Hutchinson.
File translated from T_{E}X by T_{T}H, version 2.10. On 11 Apr 1999, 23:08.

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