Page 414 - Advanced Linear Algebra

398 Advanced Linear Algebra

Thus, in both cases,

#~ ( . ( 4 4 " 4
4
and

where " 4 ~ n Ä n with  Ä

~ v Äv or " 4 " 4 ~ wÄw

depending on whether is symmetric or antisymmetric. However, in either case,
#
" are linearly independent for distinct multisets/sets 4 .
the monomials 4
Therefore, if then #~ 4 . 4 ( ( ~ for all multisets/sets 4 . Hence, if
~ and so # ~ . This shows that the restricted maps
char²-³ ~ , then 4
O O :; ²= ³ and (; ²= ³ are isomorphisms.

Theorem 14.16 Let be a finite-dimensional vector space over a field with
-
=
char²-³ ~ .
)

1 The symmetric tensor space :; ²= ³ is isomorphic to the algebra
-´ Á Ã Á µ of homogeneous polynomials, via the isomorphism

4 ÁÃÁ n Än 5 ~ ÁÃÁ ² vÄv ³

)
2 For  , the antisymmetric tensor space (; ²= ³ is isomorphic to the
c
algebra -´ Á Ã Á µ of anticommutative homogeneous polynomials of

degree , via the isomorphism

4 ÁÃÁ n Än 5 ~ ÁÃÁ ² wÄw ³

The direct sum
B

:;²= ³ ~ :; ²= ³ š -´ Á à Á µ

~
is called the symmetric tensor algebra of and the direct sum
=

c
(;²= ³ ~ (; ²= ³ š - ´ Á à Á µ

~
is called the antisymmetric tensor algebra or the exterior algebra of . These
=
vector spaces are graded algebras, where the product is defined using the vector
space isomorphisms described in the previous theorem to move the products of
-´ Á Ã Á µ and - ´ Á Ã Á µ to :;²= ³ and (;²= ³, respectively.
c

Thus, restricting the domains of the maps gives a nice description of the

symmetric and antisymmetric tensor algebras, when char²-³ ~ . However,
there are many important fields, such as finite fields, that have nonzero
characteristic. We can proceed in a different, albeit somewhat less appealing,