Double cosets are relative positions

The goal of this post is to explain something that the cool kids all understood ages ago (David Speyer, John Baez) but that I hadn’t internalized until recently.

Let be a group and let and be transitive -sets, so and for some subgroups of . In “geometric” situations (in the sense of the Erlangen program), is the symmetry group of some kind of geometry (for example, affine geometry, or Euclidean geometry), and and are spaces of “figures” in the geometry (for example, points, lines, or triangles). We’ll call the points of “-figures” and similarly for .

Now, figures in a geometry can be in various “relative positions” (or “incidence relations”) with respect to each other: for example, a point can be contained in a line, or two lines can intersect at right angles. What makes these geometrically meaningful is that they are invariant under the symmetry group of the geometry: for example, the condition that a point is contained in a line is invariant under affine symmetries, and the condition that two lines intersect at right angles is invariant under Euclidean symmetries. This motivates the following.

Definition: A relative position of -figures and -figures is a -invariant subset of , or equivalently a -invariant relation .

Any -invariant subset of decomposes into a disjoint union of -orbits: these are the atomic relative positions.

Proposition: -orbits of the action of on (equivalently, the atomic relative positions of -figures and -figures) can canonically be identified with double cosets , via the map

where means the image of under .

This is the conceptual interpretation of double cosets. It took an annoyingly long time between the first time I was introduced to double cosets (which I believe was in 2010) and the time I internalized the above fact (which was this year, 2015). Unlike the usual definition, this interpretation naturally generalizes to a notion of “triple cosets” (-orbits on a triple product ), and so forth.

Example. Let be the group of isometries of Euclidean space, which more explicitly is the semidirect product . If are both the -space of points in , then the atomic relative positions have the form “a point has distance from another point,” where is any nonnegative real.

Example. Let be the general linear group over a field and let be the Borel subgroup of upper triangular matrices. is the space of complete flags in . As it turns out, there are exactly atomic relative positions of a pair of complete flags. When is a finite field these form a basis of a Hecke algebra. In general they label the Bruhat decomposition of .

For example, when , a complete flag is just a line in , and there are two atomic relative positions: the lines can be identical or they can be different. When , a complete flag is a line contained in a plane in , and there are six atomic relative positions. Letting denote a second complete flag, they are

• ,
• ,
• ,
• ,
• ,
• .

Instead of thinking about relations as conditions on a pair of complete flags, we can also think about them as partial multi-valued functions from complete flags to complete flags. In those terms the six atomic relative positions are

• Do nothing,
• Pick a different plane,
• Pick a different line,
• Pick a different plane still containing the original line, then pick a different line not contained in the original plane,
• Pick a different plane not containing the original line, then pick a different line contained in the original plane,
• Pick a different plane not containing the original line, then pick a different line not contained in the original plane.