“Symmetric power”的意思、由来-开放百科全书

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Symmetric power

释义

In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product by the permutation action of the symmetric group .

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 , 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 is an affine variety, then the GIT quotient is the n-th symmetric power of X.

References

{{Citation | last1=Eisenbud | first1=David | authorlink1=David Eisenbud| last2=Harris | first2=Joe | authorlink2=Joe Harris (mathematician)| title=3264 and All That: A Second Course in Algebraic Geometry }}

{{cite web|last=Hopkins|first=Michael J.|authorlink=Michael J. Hopkins|url=http://www.math.harvard.edu/~lurie/ThursdayFall2017/Lecture13-Symmetric-power.pdf|title=Symmetric powers of the sphere}}

1 : Symmetric relations

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