# 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 , ỳ, vr Î V :         T(vc(1) , ỳ, vc(r))
 =
 e(c) T(v1 , ỳ, vr )  .
If a tensor T has a symmetry (C , e) and T is an isomer
 Tḃ(v1 , ỳ, vr )
 =
 T(vp(1) , ỳ, vp(r))
of T defined by a fixed permutation p Î Sr, then T possesses the symmetry (C, e) which is given by
 Cḃ   =   p ḞC Ḟp-1
 ,
 "cḃ Î Cḃ:     eḃ(cḃ)    =   e(p-1 ḞcḃḞp)  .
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.