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
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 C   ,  "v1 , , vr V :         T(vc(1) , , vc(r))
e(c) T(v1 , , vr )  .
If a tensor T has a symmetry (C , e) and T is an isomer
T(v1 , , vr )
T(vp(1) , , vp(r))
of T defined by a fixed permutation p Sr, then T possesses the symmetry (C, e) which is given by
C   =   p C p-1
"c C:     e(c)    =   e(p-1 cp)  .
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.

Back to my home page
Bernd Fiedler , 11.04.1999

File translated from TEX by TTH, version 2.10.
On 11 Apr 1999, 23:08.
WEBCounter by GOWEB WEBCounter by GOWEB
Locations of visitors to this page visitors since 20. April 2003