Symmetric power

Template:One source Template:Unfocused In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product $X^n:=X\times \cdots \times X$ 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 $X=\mathrm{Spec}(A)$ is an affine variety, then the GIT quotient $\mathrm{Spec}((A{\otimes }_k\dots {\otimes }_{k}A{)}^{{𝔖}_n})$ is the n-th symmetric power of X.

References

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External links

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