Equivariant map

In

Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant.

In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and equivariant stable homotopy theory.

Examples

Elementary geometry

In the geometry of

The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However, these changes happen in a predictable way: if a triangle is scaled by a factor of s, the perimeter also scales by s and the area scales by s². In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.

Statistics

Another class of simple examples comes from

The median of a sample is equivariant for a much larger group of transformations, the (strictly) monotonic functions of the real numbers. This analysis indicates that the median is more robust against certain kinds of changes to a data set, and that (unlike the mean) it is meaningful for ordinal data.^[3]

The concepts of an invariant estimator and equivariant estimator have been used to formalize this style of analysis.

Representation theory

In the

Under some conditions, if X and Y are both

for all g ∈ G and all x in X.[6]

If one or both of the actions are right actions the equivariance condition may be suitably modified:

f(x·g) = f(x)·g; (right-right)

f(x·g) = g⁻¹·f(x); (right-left)

f(g·x) = f(x)·g⁻¹; (left-right)

Equivariant maps are

The equivariance condition can also be understood as the following commutative diagram. Note that ${\displaystyle g\cdot }$ denotes the map that takes an element ${\displaystyle z}$ and returns ${\displaystyle g\cdot z}$.

Generalization

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Equivariant maps can be generalized to arbitrary

Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ. Using natural transformations as morphisms, one can form the category of all representations of G in C. This is just the functor category C^G.

For another example, take C = Top, the category of topological spaces. A representation of G in Top is a topological space on which G acts continuously. An equivariant map is then a continuous map f : X → Y between representations which commutes with the action of G.

See also

• cellular automata
  in terms of equivariant maps

References