Symmetric power

In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product $$X^n:=X \times \cdots \times X$$ by the permutation action of the symmetric group $$\mathfrak{S}_n$$.

More precisely, the notion exists at least in the following three areas:
 * In linear algebra, the n-th symmetric power of a vector space V is the vector subspace of the symmetric algebra of V consisting of degree-n elements (here the product is a tensor product).
 * In algebraic topology, the n-th symmetric power of a topological space X is the quotient space $$X^n/\mathfrak{S}_n$$, as in the beginning of this article.
 * In algebraic geometry, a symmetric power is defined in a way similar to that in algebraic topology. For example, if $$X = \operatorname{Spec}(A)$$ is an affine variety, then the GIT quotient $$\operatorname{Spec}((A \otimes_k \dots \otimes_k A)^{\mathfrak{S}_n})$$ is the n-th symmetric power of X.