Equivariant

In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant if

Related Topics:
Mathematics - Function - Set - Action of a group - Group

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:f(g·x) = g·f(x)

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for all g ∈ G and all x in X. Note that if one or both of the actions are on the right the equivariance condition must be suitably modified:

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:f(x·g) = f(x)·g

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:f(x·g) = g−1·f(x)

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:f(g·x) = f(x)·g−1

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Equivariant maps are homomorphisms in the category of G-sets (for a fixed G). Hence they are also known as G-maps or G-homomorphisms. Isomorphisms of G-sets are simply bijective equivariant maps.

Related Topics:
Homomorphism - Category - Isomorphism - Bijective

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The equivariance condition can also be understood as the following commutative diagram. Note that gcdot denotes the map that takes an element z and returns gcdot z.

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