Double Coset

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. More precisely, let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in , the -double coset of is the set
:$HxK = \.$
When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation
: if and only if there exist in and in such that .
The set of all double cosets is denoted by $H \,\backslash G / K.$

Properties

Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and 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 and are either disjoint or identical.
* is the disjoint union of its double cosets.
* There is a one-to-one correspondence between the two double coset spaces and given by identifying with .
* If , then . If , then .
* A double coset is a union of right cosets of and left cosets of ; specifically,
*:$\begin HxK &= \bigcup_ Hxk = \coprod_ Hxk, \\ HxK &= \bigcup_ hxK = \coprod_ hxK. \end$
* The set of -double cosets is in bijection with the orbits , and also with the orbits under the mappings $HgK \to H(gK)$ and $HgK \to (Hg)K$ respectively.
* If is normal, then is a group, and the right action of on this group factors through the right action of . It follows that . Similarly, if is normal, then .
* If is a normal subgroup of , then the -double cosets are in one-to-one correspondence with the left (and right) -cosets.
* Consider as the union of a -orbit of right -cosets. The stabilizer of the right -coset with respect to the right action of is . Similarly, the stabilizer of the left -coset with respect to the left action of is .
* It follows that the number of right cosets of contained in is the index and the number of left cosets of contained in is the index . Therefore
*:\begin
, HxK, &= : H \cap xKx^, K, = , H, : K \cap x^Hx \\
\left : H\right&= \sum_ : K \cap x^Hx \\
\left : K\right&= \sum_ : H \cap xKx^
\end
*If , , and are finite, then it also follows that
*:\begin
, HxK, &= \frac = \frac, \\
\left : H\right&= \sum_ \frac, \\
\left : K\right&= \sum_ \frac.
\end
* Fix in , and let denote the double stabilizer . Then the double stabilizer is a subgroup of .
* Because is a group, for each in there is precisely one in such that , namely ; however, may not be in . Similarly, for each in there is precisely one in such that , but may not be in . The double stabilizer therefore has the descriptions
*:$(H \times K)_x = \ \cap H \times K = \ \cap H \times K.$
* ( Orbit–stabilizer theorem) There is a bijection between and under which corresponds to . It follows that if , , and are finite, then
*:, HxK, = \times K : (H \times K)_x= , H \times K, / , (H \times K)_x, .
* ( Cauchy–Frobenius lemma) Let denote the elements fixed by the action of . Then
*:$, H \,\backslash G / K, = \frac\sum_ , G^, .$
* In particular, if , , and 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 and both act by right multiplication on . Then acts by left multiplication on the product of coset spaces . The orbits of this action are in one-to-one correspondence with . This correspondence identifies with the double coset . Briefly, this is because every -orbit admits representatives of the form , and the representative is determined only up to left multiplication by an element of . Similarly, acts by right multiplication on , and the orbits of this action are in one-to-one correspondence with the double cosets . Conceptually, this identifies the double coset space with the space of relative configurations of an -coset and a -coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups , the space of -multicosets is the set of -orbits of .

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 . For example, let be the symmetric group on three letters, and let and be the cyclic subgroups generated by the transpositions and , respectively. If denotes the identity permutation, then
:$HeK = HK = \.$
This has four elements, and four does not divide six, the order of . It is also false that different double cosets have the same size. Continuing the same example,
:$H(2 3)K = \,$
which has two elements, not four.

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

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

Examples

* Let be the symmetric group, considered as permutations of the set . Consider the subgroup which stabilizes . Then consists of two double cosets. One of these is , and the other is for any permutation which does not fix . This is contrasted with , which has $n$ elements $\gamma_1 S_, \gamma_2 S_, ..., \gamma_n S_$, where each $\gamma_i(n) = i$.
* Let be the group , and let be the subgroup of upper triangular matrices. The double coset space is the Bruhat decomposition of . The double cosets are exactly , where ranges over all n-by-n permutation matrices. For instance, if , then
*:$B \,\backslash\! \operatorname_2(\mathbf) / B = \left\.$

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

Suppose that is a group and that , , and are subgroups. Under certain finiteness conditions, there is a product on the free abelian group generated by the - and -double cosets with values in the free abelian group generated by the -double cosets. This means there is a bilinear function
:\mathbf \backslash G / K\times \mathbf \backslash G / L\to \mathbf \backslash G / L

Assume for simplicity that is finite. To define the product, reinterpret these free abelian groups in terms of the group algebra of as follows. Every element of has the form
:\sum_ f_ \cdot xK
where is a set of integers indexed by the elements of . This element may be interpreted as a -valued function on , specifically, . This function may be pulled back along the projection which sends to the double coset . This results in a function . By the way in which this function was constructed, it is left invariant under and right invariant under . The corresponding element of the group algebra is
:\sum_ f_ \cdot
and this element is invariant under left multiplication by and right multiplication by . Conceptually, this element is obtained by replacing by the elements it contains, and the finiteness of ensures that the sum is still finite. Conversely, every element of which is left invariant under and right invariant under is the pullback of a function on . Parallel statements are true for and .

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

An important special case is when . In this case, the product is a bilinear function
:\mathbf \backslash G / H\times \mathbf \backslash G / H\to \mathbf \backslash G / H
This product turns into an associative ring whose identity element is the class of the trivial double coset . In general, this ring is non-commutative. For example, if , then the ring is the group algebra , and a group algebra is a commutative ring if and only if the underlying group is abelian.

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

If is not normal, then may be commutative even if 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 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 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 is replaced by an algebra of functions such as or , 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.

Applications

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

Double cosets are important in connection with representation theory, when a representation of is used to construct an induced representation of , which is then restricted to . 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 can form a commutative ring under convolution: see Gelfand pair.

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

In number theory, the Hecke algebra corresponding to a congruence subgroup of the modular group is spanned by elements of the double coset space $\Gamma \backslash \mathrm_2^+(\mathbb) / \Gamma$; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators $T_m$ corresponding to the double cosets $\Gamma_0(N) g \Gamma_0(N)$ or $\Gamma_1(N) g \Gamma_1(N)$, where $g= \left( \begin 1 & 0 \\ 0 & m \end \right)$ (these have different properties depending on whether and are coprime or not), and the diamond operators $\langle d \rangle$ given by the double cosets $\Gamma_1(N) \left(\begin a & b \\ c & d \end \right) \Gamma_1(N)$ where $d \in (\mathbb/N\mathbb)^\times$ and we require $\left( \begin a & b \\ c & d \end \right)\in \Gamma_0(N)$ (the choice of does not affect the answer).

References

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Group theory