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 \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
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\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, Cambridge University Press, ISBN 978-1-107-01708-5
External links
- Hopkins, Michael J. (March 2018). "Symmetric powers of the sphere"