Page 410 - Advanced Linear Algebra

394 Advanced Linear Algebra

in the sum for . For example, if
#

# ~ n n b n n b n n

then

. ¸ Á Á ¹ ²#³ ~ ¸ n n Á n n ¹

#
Let :²#³ denote the sum of the terms of associated with . ²#³ . For
4
4
example,

: ¸ Á Á ¹ ²#³ ~ n n b n n

Thus, can be written in the form
#
p s
#~ : ²#³ ~ 4 ! !
4 4 q !. ²#³ t
4
where the sum is over a collection of multisets 4 with : ² # ³ £ 4 . Note also
4 £
that ! since !. ²#³ . Finally, let
" ~ 4 n Ä n
.

be the unique member of . 4 for which  Ä
Now we can get to the business at hand.
Symmetric and Antisymmetric Tensors

Let : be the symmetric group on ¸ Á à Á ¹ . For each  : , the multilinear

map ¢ = d ¦ ; ²= ³ defined by

²% Á Ã Á % ³ ~ % nÄn%

determines a unique linear operator on ;²= ³ for which
²% nÄn% ³ ~ % n Än%
For example, if ~ and ~ ² ³ , then

² ³ ²# n# n# ³ ~ # n# n#

Let ¸ ÁÃÁ ¹ be a basis for . Since is a bijection of the basis
=

8 8 ~¸ n Ä n “  ¹

it follows that is an isomorphism of ;²= ³ . Note also that is a
permutation of each . , that is, the sets . 4 4 are invariant under .
Definition Let be a finite-dimensional vector space.
=