Double coset

Template:Short description In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups, generalizing the notion of a single coset.<ref name=Hall>Template:Citation</ref><ref>Template:Citation</ref>

DefinitionEdit

Let Template:Math be a group, and let Template:Math and Template:Math be subgroups. Let Template:Math act on Template:Math by left multiplication and let Template:Math act on Template:Math by right multiplication. For each Template:Math in Template:Math, the Template:Math-double coset of Template:Math is the set

<math>HxK = \{ hxk \colon h \in H, k \in K \}.</math>

When Template:Math, this is called the Template:Math-double coset of Template:Math. Equivalently, Template:Math is the equivalence class of Template:Math under the equivalence relation

Template:Math if and only if there exist Template:Math in Template:Math and Template:Math in Template:Math such that Template:Math.

The set of all <math>(H,K)</math>-double cosets is denoted by <math>H \,\backslash G / K.</math>

PropertiesEdit

Suppose that Template:Math is a group with subgroups Template:Math and Template:Math acting by left and right multiplication, respectively. The Template:Math-double cosets of Template:Math may be equivalently described as orbits for the product group Template:Math acting on Template:Math by Template:Math. Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because Template:Math is a group and Template:Math and Template:Math are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.

Two double cosets Template:Math and Template:Math are either disjoint or identical.
Template:Math is the disjoint union of its double cosets.
There is a one-to-one correspondence between the two double coset spaces Template:Math and Template:Math given by identifying Template:Math with Template:Math.
If Template:Math, then Template:Math. If Template:Math, then Template:Math.
A double coset Template:Math is a union of right cosets of Template:Math and left cosets of Template:Math; specifically,
<math>\begin{align}

HxK &= \bigcup_{k \in K} Hxk = \coprod_{Hxk \,\in\, H \backslash HxK} Hxk, \\ HxK &= \bigcup_{h \in H} hxK = \coprod_{hxK \,\in\, HxK / K} hxK. \end{align}</math>

The set of Template:Math-double cosets is in bijection with the orbits Template:Math, and also with the orbits Template:Math under the mappings <math>HgK \to H(gK)</math> and <math>HgK \to (Hg)K</math> respectively.
If Template:Math is normal, then Template:Math is a group, and the right action of Template:Math on this group factors through the right action of Template:Math. It follows that Template:Math. Similarly, if Template:Math is normal, then Template:Math.
If Template:Math is a normal subgroup of Template:Math, then the Template:Math-double cosets are in one-to-one correspondence with the left (and right) Template:Math-cosets.
Consider Template:Math as the union of a Template:Math-orbit of right Template:Math-cosets. The stabilizer of the right Template:Math-coset Template:Math with respect to the right action of Template:Math is Template:Math. Similarly, the stabilizer of the left Template:Math-coset Template:Math with respect to the left action of Template:Math is Template:Math.
It follows that the number of right cosets of Template:Math contained in Template:Math is the index Template:Math and the number of left cosets of Template:Math contained in Template:Math is the index Template:Math. Therefore
<math>\begin{align}

|HxK| &= [H : H \cap xKx^{-1}] |K| = |H| [K : K \cap x^{-1}Hx], \\

\left[G : H\right] &= \sum_{HxK \,\in\, H \backslash G / K} [K : K \cap x^{-1}Hx], \\
\left[G : K\right] &= \sum_{HxK \,\in\, H \backslash G / K} [H : H \cap xKx^{-1}].

\end{align}</math>

If Template:Math, Template:Math, and Template:Math are finite, then it also follows that
<math>\begin{align}

|HxK| &= \frac{|H||K|}{|H \cap xKx^{-1}|} = \frac{|H||K|}{|K \cap x^{-1}Hx|}, \\

\left[G : H\right] &= \sum_{HxK \,\in\, H \backslash G / K} \frac{|K|}{|K \cap x^{-1}Hx|}, \\
\left[G : K\right] &= \sum_{HxK \,\in\, H \backslash G / K} \frac{|H|}{|H \cap xKx^{-1}|}.

\end{align}</math>

Fix Template:Math in Template:Math, and let Template:Math denote the double stabilizer Template:Math}. Then the double stabilizer is a subgroup of Template:Math.
Because Template:Math is a group, for each Template:Math in Template:Math there is precisely one Template:Math in Template:Math such that Template:Math, namely Template:Math; however, Template:Math may not be in Template:Math. Similarly, for each Template:Math in Template:Math there is precisely one Template:Math in Template:Math such that Template:Math, but Template:Math may not be in Template:Math. The double stabilizer therefore has the descriptions
<math>(H \times K)_x = \{(h, x^{-1}h^{-1}x) \colon h \in H\} \cap H \times K = \{(xk^{-1}x^{-1}, k) \colon k \in K\} \cap H \times K.</math>
(Orbit–stabilizer theorem) There is a bijection between Template:Math and Template:Math under which Template:Math corresponds to Template:Math. It follows that if Template:Math, Template:Math, and Template:Math are finite, then
<math>|HxK| = [H \times K : (H \times K)_x] = |H \times K| / |(H \times K)_x|.</math>
(Cauchy–Frobenius lemma) Let Template:Math denote the elements fixed by the action of Template:Math. Then
<math>|H \,\backslash G / K| = \frac{1}{|H||K|}\sum_{(h, k) \in H \times K} |G^{(h, k)}|.</math>
In particular, if Template:Math, Template:Math, and Template:Math are finite, then the number of double cosets equals the average number of points fixed per pair of group elements.

There is an equivalent description of double cosets in terms of single cosets. Let Template:Math and Template:Math both act by right multiplication on Template:Math. Then Template:Math acts by left multiplication on the product of coset spaces Template:Math. The orbits of this action are in one-to-one correspondence with Template:Math. This correspondence identifies Template:Math with the double coset Template:Math. Briefly, this is because every Template:Math-orbit admits representatives of the form Template:Math, and the representative Template:Math is determined only up to left multiplication by an element of Template:Math. Similarly, Template:Math acts by right multiplication on Template:Math, and the orbits of this action are in one-to-one correspondence with the double cosets Template:Math. Conceptually, this identifies the double coset space Template:Math with the space of relative configurations of an Template:Math-coset and a Template:Math-coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups Template:Math, the space of Template:Math-multicosets is the set of Template:Math-orbits of Template:Math.

The analog of Lagrange's theorem for double cosets is false. This means that the size of a double coset need not divide the order of Template:Math. For example, let Template:Math be the symmetric group on three letters, and let Template:Math and Template:Math be the cyclic subgroups generated by the transpositions Template:Math and Template:Math, respectively. If Template:Math denotes the identity permutation, then

This has four elements, and four does not divide six, the order of Template:Math. It is also false that different double cosets have the same size. Continuing the same example,

which has two elements, not four.

However, suppose that Template:Math is normal. As noted earlier, in this case the double coset space equals the left coset space Template:Math. Similarly, if Template:Math is normal, then Template:Math is the right coset space Template:Math. Standard results about left and right coset spaces then imply the following facts.

Template:Math for all Template:Math in Template:Math. That is, all double cosets have the same cardinality.
If Template:Math is finite, then Template:Math. In particular, Template:Math and Template:Math divide Template:Math.

ExamplesEdit

Let Template:Math be the symmetric group, considered as permutations of the set Template:Math}. Consider the subgroup Template:Math which stabilizes Template:Math. Then Template:Math consists of two double cosets. One of these is Template:Math, and the other is Template:Math for any permutation Template:Math which does not fix Template:Math. This is contrasted with Template:Math, which has <math>n</math> elements <math>\gamma_1 S_{n-1}, \gamma_2 S_{n-1}, ..., \gamma_n S_{n-1}</math>, where each <math>\gamma_i(n) = i</math>.
Let Template:Math be the group Template:Math, and let Template:Math be the subgroup of upper triangular matrices. The double coset space Template:Math is the Bruhat decomposition of Template:Math. The double cosets are exactly Template:Math, where Template:Math ranges over all n-by-n permutation matrices. For instance, if Template:Math, then
<math>B \,\backslash\! \operatorname{GL}_2(\mathbf{R}) / B = \left\{ B\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}B,\ B\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}B \right\}.</math>

Products in the free abelian group on the set of double cosetsEdit

Suppose that Template:Math is a group and that Template:Math, Template:Math, and Template:Math are subgroups. Under certain finiteness conditions, there is a product on the free abelian group generated by the Template:Math- and Template:Math-double cosets with values in the free abelian group generated by the Template:Math-double cosets. This means there is a bilinear function

<math>\mathbf{Z}[H \backslash G / K] \times \mathbf{Z}[K \backslash G / L] \to \mathbf{Z}[H \backslash G / L].</math>

Assume for simplicity that Template:Math is finite. To define the product, reinterpret these free abelian groups in terms of the group algebra of Template:Math as follows. Every element of Template:Math has the form

<math>\sum_{HxK \in H \backslash G / K} f_{HxK} \cdot [HxK],</math>

where Template:Math is a set of integers indexed by the elements of Template:Math. This element may be interpreted as a Template:Math-valued function on Template:Math, specifically, Template:Math. This function may be pulled back along the projection Template:Math which sends Template:Math to the double coset Template:Math. This results in a function Template:Math. By the way in which this function was constructed, it is left invariant under Template:Math and right invariant under Template:Math. The corresponding element of the group algebra Template:Math is

and this element is invariant under left multiplication by Template:Math and right multiplication by Template:Math. Conceptually, this element is obtained by replacing Template:Math by the elements it contains, and the finiteness of Template:Math ensures that the sum is still finite. Conversely, every element of Template:Math which is left invariant under Template:Math and right invariant under Template:Math is the pullback of a function on Template:Math. Parallel statements are true for Template:Math and Template:Math.

When elements of Template:Math, Template:Math, and Template:Math are interpreted as invariant elements of Template:Math, then the product whose existence was asserted above is precisely the multiplication in Template:Math. Indeed, it is trivial to check that the product of a left-Template:Math-invariant element and a right-Template:Math-invariant element continues to be left-Template:Math-invariant and right-Template:Math-invariant. The bilinearity of the product follows immediately from the bilinearity of multiplication in Template:Math. It also follows that if Template:Math is a fourth subgroup of Template:Math, then the product of Template:Math-, Template:Math-, and Template:Math-double cosets is associative. Because the product in Template:Math corresponds to convolution of functions on Template:Math, this product is sometimes called the convolution product.

An important special case is when Template:Math. In this case, the product is a bilinear function

<math>\mathbf{Z}[H \backslash G / H] \times \mathbf{Z}[H \backslash G / H] \to \mathbf{Z}[H \backslash G / H].</math>

This product turns Template:Math into an associative ring whose identity element is the class of the trivial double coset Template:Math. In general, this ring is non-commutative. For example, if Template:Math, then the ring is the group algebra Template:Math, and a group algebra is a commutative ring if and only if the underlying group is abelian.

If Template:Math is normal, so that the Template:Math-double cosets are the same as the elements of the quotient group Template:Math, then the product on Template:Math is the product in the group algebra Template:Math. In particular, it is the usual convolution of functions on Template:Math. In this case, the ring is commutative if and only if Template:Math is abelian, or equivalently, if and only if Template:Math contains the commutator subgroup of Template:Math.

If Template:Math is not normal, then Template:Math may be commutative even if Template:Math is non-abelian. A classical example is the product of two Hecke operators. This is the product in the Hecke algebra, which is commutative even though the group Template:Math is the modular group, which is non-abelian, and the subgroup is an arithmetic subgroup and in particular does not contain the commutator subgroup. Commutativity of the convolution product is closely tied to Gelfand pairs.

When the group Template:Math is a topological group, it is possible to weaken the assumption that the number of left and right cosets in each double coset is finite. The group algebra Template:Math is replaced by an algebra of functions such as Template:Math or Template:Math, and the sums are replaced by integrals. The product still corresponds to convolution. For instance, this happens for the Hecke algebra of a locally compact group.

ApplicationsEdit

When a group <math>G </math> has a transitive group action on a set <math>S</math>, computing certain double coset decompositions of <math>G </math> reveals extra information about structure of the action of <math>G </math> on <math>S </math>. Specifically, if <math>H </math> is the stabilizer subgroup of some element <math>s\in S </math>, then <math>G </math> decomposes as exactly two double cosets of <math>(H,H) </math> if and only if <math>G </math> acts transitively on the set of distinct pairs of <math>S</math>. See 2-transitive groups for more information about this action.

Double cosets are important in connection with representation theory, when a representation of Template:Math is used to construct an induced representation of Template:Math, which is then restricted to Template:Math. The corresponding double coset structure carries information about how the resulting representation decomposes. In the case of finite groups, this is Mackey's decomposition theorem.

They are also important in functional analysis, where in some important cases functions left-invariant and right-invariant by a subgroup Template:Math can form a commutative ring under convolution: see Gelfand pair.

In geometry, a Clifford–Klein form is a double coset space Template:Math, where Template:Math is a reductive Lie group, Template:Math is a closed subgroup, and Template:Math is a discrete subgroup (of Template:Math) that acts properly discontinuously on the homogeneous space Template:Math.

In number theory, the Hecke algebra corresponding to a congruence subgroup Template:Math of the modular group is spanned by elements of the double coset space <math>\Gamma \backslash \mathrm{GL}_2^+(\mathbb{Q}) / \Gamma</math>; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators <math>T_m</math> corresponding to the double cosets <math>\Gamma_0(N) g \Gamma_0(N)</math> or <math>\Gamma_1(N) g \Gamma_1(N)</math>, where <math>g= \left( \begin{smallmatrix} 1 & 0 \\ 0 & m \end{smallmatrix} \right)</math> (these have different properties depending on whether Template:Math and Template:Math are coprime or not), and the diamond operators <math> \langle d \rangle</math> given by the double cosets <math> \Gamma_1(N) \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \Gamma_1(N)</math> where <math> d \in (\mathbb{Z}/N\mathbb{Z})^\times</math> and we require <math> \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)\in \Gamma_0(N)</math> (the choice of Template:Math does not affect the answer).

ReferencesEdit

Template:Reflist