Symmetric power

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

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 ${\displaystyle X^{n}/{\mathfrak {S}}_{n}}$, 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 ${\displaystyle X=\operatorname {Spec} (A)}$ is an affine variety, then the GIT quotient ${\displaystyle \operatorname {Spec} ((A\otimes _{k}\dots \otimes _{k}A)^{{\mathfrak {S}}_{n}})}$ is the n-th symmetric power of X.