The cube and the octahedron form a dual pair.

The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout.

As well as identity can be generalized for different reductive dual pairs.

Many other matroid families come in dual pairs.

The regular polyhedra form dual pairs, with the tetrahedron being self-dual.

Given two dual pairs separated locally convex spaces and .

These occur as dual pairs in the same way as the original Platonic solids do.

Howe correspondence between representations of elements of a reductive dual pair.

Each of these forms the common core of a dual pair of regular polyhedra.

The eight dual pairs are as follows:

Let and be dual pairs.

Geometrically they come in dual pairs - one elongated, and one correspondingly squashed.

A reductive dual pair is called reducible if it can be obtained in this fashion from smaller groups, and irreducible otherwise.

Two are based on dual pairs of regular Kepler-Poinsot solids, in the same way as for the convex examples.

These occur as dual pairs as follows:

Thus, types of configurations come in dual pairs, except when taking the dual results in an isomorphic configuration.

Several classes of reductive dual pairs had appeared earlier in the work of André Weil.

If we consider a monoidal category as a bicategory with one object, a dual pair is exactly an adjoint pair.

A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed by a bilinear form.

The different dual topologies for a given dual pair are characterized by the Mackey-Arens theorem.

The Kepler-Poinsot polyhedra exist in dual pairs:

Note that X is an invariant lagrangian subspace, hence this dual pair is of type II.

Theorem (by Mackey): Given a dual pair, the bounded sets under any dual topology are identical.

In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.

These two constructions produce all irreducible reductive dual pairs over an algebraically closed field F, such as the field C of complex numbers.