Page 417 - Advanced Linear Algebra

Tensor Products 401

)
1 The dimension of the symmetric tensor space :; ²= ³ is equal to the

number of monomials of degree in the variables ÁÃÁ and this is
b c

dim²:; ²= ³³ ~ 6 7

)
2 The dimension of the exterior tensor space ²= ³ is equal to the number of

words of length in ascending order over the alphabet , ~ ¸ Á Ã Á ¹
and this is

dim ² ² = ³ ³ ~ 67

Proof. For part 1), the dimension is equal to the number of multisets of size
taken from an underlying set ¸ ÁÃÁ ¹ of size . Such multisets correspond

bijectively to the solutions, in nonnegative integers, of the equation
%b Ä b % ~

where % is the multiplicity of in the multiset. To count the number of
solutions, invent two symbols and . Then any solution % ~ to the
°
%
°
%

previous equation can be described by a sequence of 's and 's consisting of
% °'s followed by one , followed by 's and another , and so on. For example,

%
°
if ~ and ~ , the solution b b b ~ corresponds to the sequence
%%%°%°°%%
Thus, the solutions correspond bijectively to sequences consisting of 's and
%
c °'s. To count the number of such sequences, note that such a sequence can

be formed by considering b c “blanks” and selecting of these blanks for

the 's. This can be done in
%
b c
6 7

ways.
The Universal Property
We defined tensor products through a universal property, which as we have seen
is a powerful technique for determining the properties of tensor products. It is
easy to show that the symmetric tensor spaces are universal for symmetric
multilinear maps and the antisymmetric tensor spaces are universal for
antisymmetric multilinear maps.
Theorem 14.19 Let = be a finite-dimensional vector space with basis
¸ ÁÃÁ ¹.

)
1 The pair ²- ´% ÁÃÁ% µÁ !³ , where !¢ = d ¦ - ´% Á ÃÁ% µ is the

multilinear map defined by