Page 407 - Advanced Linear Algebra

Tensor Products 391

)

3 A multilinear map ¢ = ¦ > is alternate or alternating if
for some £ ¬ ²# ÁÃÁ# ³ ~
# ~ #
As in the case of bilinear forms, we have some relationships between these
concepts. In particular, if char²-³ ~ , then

alternate ¬ symmetric ¯ skew-symmetric
and if char²-³ £ , then
alternate ¯ skew-symmetric
A few remarks about permutations are in order. A permutation of the set
(
5 ~ ¸ Á Ã Á ¹ is a bijective function ¢ 5 ¦ 5. We denote the group under
) . This is the symmetric group on
:
composition of all such permutations by
symbols. A cycle of length is a permutation of the form ² Á Á Ã Á ³ , which

sends to " " b for ~ " Á Ã Á c and also sends to . All other elements

of 5 are left fixed. Every permutation is the product (composition) of disjoint
cycles.
A transposition is a cycle ² Á ³ of length . Every cycle and therefore every
(

permutation is the product of transpositions. In general, a permutation can be
)
expressed as a product of transpositions in many ways. However, no matter how
one represents a given permutation as such a product, the number of
transpositions is either always even or always odd. Therefore, we can define the
to be the parity of the number of transpositions
parity of a permutation :
in any decomposition of as a product of transpositions. The sign of a

permutation is defined by
has even parity

sg²³ ~ F
c has odd parity

If sg²³ ~ , then is an even permutation and if sg²³ ~ c , then is an

odd permutation. The sign of is often written ²c ³ .
With these facts in mind, it is apparent that is symmetric if and only if

²# Á ÃÁ# ³ ~ ²# ² ÁÃÁ# ² 1) ³ ³
and that is skew-symmetric if and only if
for all permutations :

²# Á ÃÁ# ³ ~ ²c ³ ²# ² ÁÃÁ# ² 1) ³ ³
.
for all permutations :
A word of caution is in order with respect to the notation above, which is very
convenient albeit somewhat prone to confusion. It is intended that a permutation
permutes the coordinate positions in , not the indices (despite appearances).

Suppose, for example, that ¢ s d s ¦ ? and that ¸ Á ¹ is a basis for s .