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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

\mathfrak_n

. More precisely, the notion exists at least in the following three areas: *In

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, the ''n''-th symmetric power of a vector space ''V'' is the vector subspace of the

symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...

of ''V'' consisting of degree-''n'' elements (here the product is a

tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

). *In

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, the ''n''-th symmetric power of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

''X'' is the quotient space

X^n/\mathfrak_n

, as in the beginning of this article. *In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a symmetric power is defined in a way similar to that in algebraic topology. For example, if

X = \operatorname(A)

is an

affine variety In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

, then the GIT quotient

\operatorname((A \otimes_k \dots \otimes_k A)^)

is the ''n''-th symmetric power of ''X''.

References

External links

* Symmetric relations {{math-stub