Symmetric power

Template:Short description Template:One source Template:Unfocused In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product $X^{n} := X \times \dots \times X$ by the permutation action of the symmetric group $𝔖_{n}$ .

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 $X^{n} / 𝔖_{n}$ , 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 $X = Spec (A)$ is an affine variety, then the GIT quotient $Spec ((A \otimes_{k} \dots \otimes_{k} A)^{𝔖_{n}})$ is the n-th symmetric power of X.

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