In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product X n := X × ⋯ × X {\displaystyle X^{n}:=X\times \cdots \times X} by the permutation action of the symmetric group S n {\displaystyle {\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 / S n {\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 = Spec ( A ) {\displaystyle X=\operatorname {Spec} (A)} is an affine variety, then the GIT quotient Spec ( ( A ⊗ k ⋯ ⊗ k A ) S n ) {\displaystyle \operatorname {Spec} ((A\otimes _{k}\dots \otimes _{k}A)^{{\mathfrak {S}}_{n}})} is the n-th symmetric power of X.
References
External links Hopkins, Michael J. (March 2018). "Symmetric powers of the sphere" (PDF).