Symmetric power explained

In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product

X ^{n:=X} x … x X by the permutation action of the symmetric group ak{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).

X ^{n/ ak{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 ⊗ _{k} ... ⊗ _{k} A) ^{ak{S} _{n})} is the n-th symmetric power of X.

External links

• Web site: Hopkins. Michael J.. Michael J. Hopkins. Symmetric powers of the sphere . March 2018.