Page 413 - Advanced Linear Algebra

Tensor Products 397

#~ 8 ! 9 4
4 !. 4
where 4 is a multiset and the antisymmetric tensors are

#~ 8 ²c ³ "Á! ! 9 4
4 !. 4
where 4 is a set.

We can simplify these expressions considerably by representing the inside sums
more succinctly. In the symmetric case, define a surjective linear map

¢; ²= ³ ¦ - ´ Á ÃÁ µ

by

² nÄn ³ ~ v Äv
to the same
and extending by linearity. Since takes every member of . 4
, we have
monomial "~ v Ä v , where Ä
#~ 8 8 ! 9 4 9 ~ 4 ( 4 . ( " 4
4 !. 4 4
In the antisymmetric case, define a surjective linear map
c

¢; ²= ³ ¦ - ´ Á ÃÁ µ

by

² nÄn ³ ~ w Äw
and extending by linearity. Since
"Á! ! ~ ²c ³ " 4

we have

8 #~ ²c ³ "Á! ! 9 4
4 !. 4

8 ~ " 4 9 4
4 !. 4
~ . 4 ( ( " 4 4
4