Symmetries

Tensors and symbols may have symmetry properties with respect to their indices or to some groups of their indices. Let us assume that the tensors $A$_ij, $B$_ij, $T$_ijklm, $R$_ij^{\>\> \>\> kl} and $S$_ijklmn possess the following symmetry properties

$A$_ij = A_(ij) B_ij = B_(ij) T_ijklm = T_(ijk)lm=T_ijk(lm) R_ij^{\>\> \>\> kl} = R_(ij)^{\>\> \>\> \>\> kl}=R_ij^{\>\> \>\> [kl]} S_ijklmn = - S_klijmn=- S_ijmnkl

where the standard notation was applied[2]. These symmetry properties are given in the system by the SYMMETRY command as follows

The objects with the given symmetry properties are on the left hand side of the sign => and separated by commas, while the symmetry properties are on the right hand side. The totally symmetric (antisymmetric) property is denoted simply by the keyword SYMMETRIC (ANTISYMMETRIC). If the symmetry property is valid only for certain indices or certain groups of indices then after the keywords SYMMETRIC IN (ANTISYMMETRIC IN) the indices or index groups follow in braces separated by commas. The different symmetry definitions are also separated by commas. The names of the indices are arbitrary but the positions distinguish tensors as well.

Naturally if metric tensor is introduced (see page ) then because of the one-to-one correspondence established by the metric tensor between the vector spaces and their dual spaces, the symmetry properties are inherited between the tensors. For example if $S$ is a tensor identifier then from the defintion

follows that

$S$_ij = S_ji S_i^{\>\> j} = g^jk S_ik= g^jk S_ki=S^j_{\>\> i} S^ij = g^ik S_k^{\>\> j}= g^ik S^j_{\>\> k}=S^ji.

When evaluating tensor expressions the program uses the symmetry properties and the inherited symmetry properties of the objects too. It does not consist of a general discrete group analysis - that would not be a simple problem itself and would require much computation time - but the simplest and most frequently used relations will be applied: if $S$_..ij.. is symmetric, $A..ij..$ is antisymmetric with respect to the indices $i,j$, $Ri$, $Ti$ and $T..ij..$ are arbitrary indexed objects then the program applies the following type of identities and their combinations:

$S$_..ij.. A^..ij.. = 0 T_i T_j A^..ij.. = 0 S_..ij.. (T^..ij..-T^..ji..) = 0 S_..ij.. (Tⁱ R^j - T^j Rⁱ ) = 0

A symmetry definition regarding to an object already of symmetry properties will not overwrite but rather extends the symmetry properties after checking them. If one would like to see the symmetry properties of an object, the SYMMETRY command is able to visualize them too:

It is possible to remove the symmetry properties of the objects by the command CLEAR SYMMETRY. It removes all the symmetry properties of the given objects.

Examples: