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 S_r 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 , ź, v_r Î V : T(v_c(1) , ź, v_c(r))

e(c) T(v₁ , ź, v_r ) .

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

T˘(v₁ , ź, v_r )

T(v_p(1) , ź, 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˘ = 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 S_r with r Ł 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 T_EX by T_TH, version 2.10.
On 11 Apr 1999, 23:08.