Page 412 - Advanced Linear Algebra
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396 Advanced Linear Algebra
. ~ 4 . Z 4 r . Z 4 Z
where . Z are the tensors that have in positions and :
4
Z
. 4 ~ ¸ nÄn nÄn nÄn ¹
position position
Then ² ³ fixes each element of . 4 Z and sends the elements of . Z 4 Z to other
elements of . ZZ 4 . Hence, applying ² ³ to the corresponding decomposition of
:²#³:
4
:²#³ ~ : 4 Z b : 4 Z Z
4
gives
c²: 4 Z b : ³ ~ c: 4 Z 4 Z ~ ² ³ : 4 ~ : 4 Z b : 4² ³ Z Z
and so : Z 4 ~ , whence 4 ² # ³ ~ . Thus, 4 is a set.
:
,
Now, since for any :
. ~ 4 ¸ ! ! . ¹ 4
equation (14.4) implies that
²c ³ !~ ! ! 9 ~ ! 8 ! !~ c ! !
!. 4 !. 4 !. 4 !. 4
which holds if and only if ! ~²c ³ c ! , or equivalently,
! ! ~²c ³
for all !. 4 and : . Choosing "~" 4 ~ n Ä n , where
Ä , as standard-bearer, if " Á ! denotes the permutation for which
²"³ ~ !, then
"Á!
"Á! ! " ~²c ³
Thus, is antisymmetric if and only if it has the form
#
#~ 8 ²c ³ "Á! ! 9 4
4 !. 4
where 4 " ~ £ and the sum is over a family of sets .
In summary, the symmetric tensors are
