The 
th
symmetric power 
of a vector space 
is the quotient of 
by the relations identifying tensors that differ
by a permutation of their factors. For nonnegative
integer 
and 
,
its dimension is
 |
Over a field of characteristic zero it can also be identified with the invariant tensors by averaging over the symmetric
group. For example, the second symmetric power of a two-dimensional space has
basis 
,
, and 
. Iterated symmetric powers, which need not agree in the
two possible orders, are compared in Foulkes' conjecture.
