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

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 $G$ 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 $X$-figures and $Y$-figures is a $G$-invariant subset of $X \times Y$, or equivalently a $G$-invariant relation $R : X \to Y$.

Any $G$-invariant subset of $X \times Y$ decomposes into a disjoint union of $G$-orbits: these are the atomic relative positions.

Proposition: $G$-orbits of the action of $G$ on $G/H \times G/K$ (equivalently, the atomic relative positions of $X$-figures and $Y$-figures) can canonically be identified with double cosets $H \backslash G/K$, via the map

$\displaystyle G/H \times G/K \ni ([g_1], [g_2]) \mapsto [g_1^{-1} g_2] \in H \backslash G/K$

where $[g] \in G/H$ means the image of $g \in G$ under $G \to G/H$.

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” ($G$-orbits on a triple product $X \times Y \times Z$), and so forth.

Example. Let $G = \text{Isom}(\mathbb{R}^n)$ be the group of isometries of Euclidean space, which more explicitly is the semidirect product $\mathbb{R}^n \rtimes O(n)$. If $X = Y$ are both the $G$-space of points in $\mathbb{R}^n$, then the atomic relative positions have the form “a point has distance $r$ from another point,” where $r$ is any nonnegative real.

Example. Let $G = GL_n(k)$ be the general linear group over a field $k$ and let $H = K = B$ be the Borel subgroup of upper triangular matrices. $G/B$ is the space of complete flags in $V = k^n$. As it turns out, there are exactly $n!$ atomic relative positions of a pair of complete flags. When $k$ is a finite field these form a basis of a Hecke algebra. In general they label the Bruhat decomposition of $G$.

For example, when $n = 2$, a complete flag is just a line in $V = k^2$, and there are two atomic relative positions: the lines can be identical or they can be different. When $n = 3$, a complete flag is a line $V_1$ contained in a plane $V_2$ in $V = k^3$, and there are six atomic relative positions. Letting $W_1 \subset W_2$ denote a second complete flag, they are

• $V_1 = W_1, V_2 = W_2$,
• $V_1 = W_1, V_2 \neq W_2$,
• $V_1 \neq W_1, V_2 = W_2$,
• $V_1 \neq W_1, V_1 \subset W_2, W_1 \not\subset V_2, V_2 \neq W_2$,
• $V_1 \neq W_1, W_1 \subset V_2, V_1 \not\subset W_2, V_2 \neq W_2$,
• $V_1 \neq W_2, V_1 \not\subset W_2, W_1 \not\subset V_2, V_2 \neq W_2$.

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.

### 4 Responses

1. A familiar example is the relative position of two lattices $L,L'\simeq \mathbb{Z}^n$ in a vector space $\mathbb{Q}^n$. Focusing on one $p$ at a time we get the relative positions of two $\mathbb{Z}_p$-lattices in $\mathbb{Q}_p^n$, essentially the possible relations between vertices of the Bruhat-Tits building for $\text{GL}_n\mathbb{Q}_p$ (except we have not passed to homothety classes, though we could). The classical Hecke operators $T_{p,k}$ then just correspond to the different relative positions that are “adjacent”.

An even more familiar example is the relative position of two positive rational numbers $a$ and $b$, which is just the ratio $\frac{a}{b}$. (This is the case $n=1$ of the previous paragraph.)

2. […] « Double cosets are relative positions […]

3. I like to think of the map $\ominus$ from two G-sets X, Y to $X \times Y / G^\Delta$ defined by $x \ominus y := \{ (gx, gy): g \in G\}$ as an abstract subtraction operation; thus the abstract difference between two points in homogenous spaces is a double coset.

4. This latter example is soooo much better with pictures (in RP^2).