Commutation symmetries of tensor indices

Let us consider covariant tensors T of order r over a finite-dimensional vector space V. A commutation symmetry of such tensors is a pair (C , e) where C is a subgroup of the symmetric group Sr and e is a homomorphism
e: C
®
S1
of C onto a finite subgroup of the group S1 of complex units.
 
We say that a covariant tensor T of order r possesses a symmetry (C , e) if T fulfils
"c Î C   ,  "v1 , ¼, vr Î V :         T(vc(1) , ¼, vc(r))
=
e(c) T(v1 , ¼, vr )  .
If a tensor T has a symmetry (C , e) and T¢ is an isomer
T¢(v1 , ¼, vr )
=
T(vp(1) , ¼, vp(r))
of T defined by a fixed permutation p Î Sr, then T¢ possesses the symmetry (C¢, e¢) which is given by
C¢   =   p °C °p-1
      ,      
"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 Sr 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.
 

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Bernd Fiedler , 04.09.2022
 

File translated from TEX by TTH, version 2.10.
On 11 Apr 1999, 23:08.