Page 409 - Advanced Linear Algebra
P. 409
Tensor Products 393
We will also need the counterpart of -´ Á Ã Á µ in which multiplication acts
anticommutatively, that is, ~ c .
Definition Let , ~ ² ÁÃÁ ³ be a sequence of independent variables. For
c
, let - ´ ÁÃÁ µ be the vector space over with basis
-
²,³~¸ Ä Ä ¹
7
consisting of all words of length over that are in ascending order. Let
,
, which we identify with
-´ Á Ã Á µ ~ - c - by identifying with - .
Define a product on the direct sum
c
-´ Á Ã Á µ ~ - c
~
as follows. First, the product w of monomials ~ % Ä% - c and
~ & Ä& - is defined as follows:
c
) has a repeated factor then w ~ .
1 If %Ä% & Ä&
) , via the
2 Otherwise, reorder %Ä% & Ä& in ascending order, say ' Ä' b
permutation and set
w ~ ²c ³ ' Ä' b
c
Extend the product by distributivity to -´ Á Ã Á µ . The resulting product
)
(
c
-
makes -´ Á Ã Á µ into a noncommutative algebra over . This product is
c
called the wedge product or exterior product on -´ Á Ã Á µ .
For example, by definition of wedge product,
w w ~ c w w
Let 8 ~¸ Á Ã Á ¹ be a basis for = . It will be convenient to group the
according to their index multiset.
decomposable basis tensors n Ä n
Specifically, for each multiset 4 ~ ¸ ÁÃÁ ¹ with , let . 4 be the
set of all tensors
n Ä n
where ² ÁÃÁ ³ is a permutation of ¸ Á ÃÁ ¹ . For example, if
4 ~ ¸ Á Á ¹, then
. 4 ~ ¸ n n Á n n Á n n ¹
If #; ²= ³ has the form
#~ n Ä n
ÁÃÁ
ÁÃÁ
£ , then let . ²#³ be the subset of . whose elements appear
where ÁÃÁ 4 4
