Symmetric power

In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product [math]\displaystyle{ X^n:=X \times \cdots \times X }[/math] by the permutation action of the symmetric group [math]\displaystyle{ \mathfrak{S}_n }[/math]. 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 [math]\displaystyle{ X^n/\mathfrak{S}_n }[/math], 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 [math]\displaystyle{ X = \operatorname{Spec}(A) }[/math] is an affine variety, then the GIT quotient [math]\displaystyle{ \operatorname{Spec}((A \otimes_k \dots \otimes_k A)^{\mathfrak{S}_n}) }[/math] is the n-th symmetric power of X.

References

• Eisenbud, David; Harris, Joe, 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1-107-01708-5, https://books.google.com/books?id=au67CwAAQBAJ&pg=PR5 

External links

• Hopkins, Michael J. (March 2018). "Symmetric powers of the sphere". http://www.math.harvard.edu/~lurie/ThursdayFall2017/Lecture13-Symmetric-power.pdf.