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 \( {\mathfrak {S}}_{n} \).
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\( X=\operatorname {Spec}(A) \) 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.
References
Eisenbud, David; Harris, Joe, 3264 and All That: A Second Course in Algebraic Geometry
External links
Hopkins, Michael J. "Symmetric powers of the sphere" (PDF).
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Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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