Editing Double coset

{{Short description|Concept in math}}
In [[group theory]], a field of [[mathematics]], a '''double coset''' is a collection of [[group (mathematics)|group]] elements which are equivalent under the symmetries coming from two [[subgroup]]s, generalizing the notion of a single [[coset]].<ref name=Hall>{{citation|first=Marshall|last=Hall, Jr.|title=The Theory of Groups|place=New York|publisher=Macmillan|year=1959|pages=14–15}}</ref><ref>{{citation|first=Homer|last=Bechtell|title=The Theory of Groups|publisher=Addison-Wesley|year=1971|page=101}}</ref>  

==Definition==
Let {{math|''G''}} be a group, and let {{math|''H''}} and {{math|''K''}} be subgroups.  Let {{math|''H''}} [[group action|act]] on {{math|''G''}} by left multiplication and let {{math|''K''}} act on {{math|''G''}} by right multiplication.  For each {{math|''x''}} in {{math|''G''}}, the '''{{math|(''H'', ''K'')}}-double coset of {{math|''x''}}''' is the [[set (mathematics)|set]]
:<math>HxK = \{ hxk \colon h \in H, k \in K \}.</math>
When {{math|1=''H'' = ''K''}}, this is called the '''{{math|''H''}}-double coset of {{math|''x''}}'''.  Equivalently, {{math|''HxK''}} is the [[equivalence class]] of {{math|''x''}} under the [[equivalence relation]]
:{{math|''x'' ~ ''y''}} if and only if there exist {{math|''h''}} in {{math|''H''}} and {{math|''k''}} in {{math|''K''}} such that {{math|1=''hxk'' = ''y''}}.
The set of all <math>(H,K)</math>-double cosets is denoted by <math>H \,\backslash G / K.</math>

== Properties ==

Suppose that {{math|''G''}} is a group with subgroups {{math|''H''}} and {{math|''K''}} acting by left and right multiplication, respectively.  The {{math|(''H'', ''K'')}}-double cosets of {{math|''G''}} may be equivalently described as [[Orbit (group theory)|orbit]]s for the [[Direct product of groups|product group]] {{math|''H'' × ''K''}} acting on {{math|''G''}} by {{math|1=(''h'', ''k'')&thinsp;⋅&thinsp;''x'' = ''hxk''<sup>−1</sup>}}.  Many of the basic properties of double cosets follow immediately from the fact that they are orbits.  However, because {{math|''G''}} is a group and {{math|''H''}} and {{math|''K''}} are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary [[group action]]s, and they have additional properties that are false for more general actions.

* Two double cosets {{math|''HxK''}} and {{math|''HyK''}} are either [[disjoint sets|disjoint]] or identical.
* {{math|''G''}} is the [[disjoint union]] of its double cosets.
* There is a one-to-one correspondence between the two double coset spaces {{math|''H'' \ ''G'' / ''K''}} and {{math|''K'' \ ''G'' / ''H''}} given by identifying {{math|''HxK''}} with {{math|''Kx''<sup>−1</sup>''H''}}.
* If {{math|1=''H'' = <nowiki>{</nowiki>1<nowiki>}</nowiki>}}, then {{math|1=''H'' \ ''G'' / ''K'' = ''G'' / ''K''}}.  If {{math|1=''K'' = <nowiki>{</nowiki>1<nowiki>}</nowiki>}}, then {{math|1=''H'' \ ''G'' / ''K'' = ''H'' \ ''G''}}.
* A double coset {{math|''HxK''}} is a [[union (set theory)|union]] of right cosets of {{math|''H''}} and left cosets of {{math|''K''}}; 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 {{math|(''H'', ''K'')}}-double cosets is in [[bijection]] with the orbits {{math|''H'' \ (''G'' / ''K'')}}, and also with the orbits {{math|(''H'' \ ''G'') / ''K''}} under the mappings <math>HgK \to H(gK)</math> and <math>HgK \to (Hg)K</math> respectively.
* If {{math|''H''}} is [[normal subgroup|normal]], then {{math|''H'' \ ''G''}} is a group, and the right action of {{math|''K''}} on this group factors through the right action of {{math|''H'' \ ''HK''}}.  It follows that {{math|1=''H'' \ ''G'' / ''K'' = ''G'' / ''HK''}}.  Similarly, if {{math|''K''}} is normal, then {{math|1=''H'' \ ''G'' / ''K'' = ''HK'' \ ''G''}}.
* If {{math|''H''}} is a normal subgroup of {{math|''G''}}, then the {{math|''H''}}-double cosets are in one-to-one correspondence with the left (and right) {{math|''H''}}-cosets.
* Consider {{math|''HxK''}} as the union of a {{math|''K''}}-orbit of right {{math|''H''}}-cosets.  The stabilizer of the right {{math|''H''}}-coset {{math|''Hxk'' ∈ ''H'' \ ''HxK''}} with respect to the right action of {{math|''K''}} is {{math|''K'' ∩ (''xk'')<sup>−1</sup>''Hxk''}}.  Similarly, the stabilizer of the left {{math|''K''}}-coset {{math|''hxK'' ∈ ''HxK'' / ''K''}} with respect to the left action of {{math|''H''}} is {{math|''H'' ∩ ''hxK''(''hx'')<sup>−1</sup>}}.
* It follows that the number of right cosets of {{math|''H''}} contained in {{math|''HxK''}} is the [[index of a subgroup|index]] {{math|[''K'' : ''K'' ∩ ''x''<sup>−1</sup>''Hx'']}} and the number of left cosets of {{math|''K''}} contained in {{math|''HxK''}} is the index {{math|[''H'' : ''H'' ∩ ''xKx''<sup>−1</sup>]}}.  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 {{math|''G''}}, {{math|''H''}}, and {{math|''K''}} 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 {{math|''x''}} in {{math|''G''}}, and let {{math|(''H'' × ''K'')<sub>''x''</sub>}} denote the double stabilizer {{math|1={(''h'', ''k'') : ''hxk'' = ''x'' }}}.  Then the double stabilizer is a subgroup of {{math|''H'' × ''K''}}.
* Because {{math|''G''}} is a group, for each {{math|''h''}} in {{math|''H''}} there is precisely one {{math|''g''}} in {{math|''G''}} such that {{math|1=''hxg'' = ''x''}}, namely {{math|1=''g'' = ''x''<sup>−1</sup>''h''<sup>−1</sup>''x''}}; however, {{math|''g''}} may not be in {{math|''K''}}.  Similarly, for each {{math|''k''}} in {{math|''K''}} there is precisely one {{math|''g''&prime;}} in {{math|''G''}} such that {{math|1=''g''&prime;''xk'' = ''x''}}, but {{math|''g''&prime;}} may not be in {{math|''H''}}.  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>
* ([[Group action#Orbit-stabilizer theorem|Orbit–stabilizer theorem]]) There is a bijection between {{math|''HxK''}} and {{math|(''H'' × ''K'') / (''H'' × ''K'')<sub>''x''</sub>}} under which {{math|''hxk''}} corresponds to {{math|(''h'', ''k''<sup>−1</sup>)(''H'' × ''K'')<sub>''x''</sub>}}.  It follows that if {{math|''G''}}, {{math|''H''}}, and {{math|''K''}} are finite, then
*:<math>|HxK| = [H \times K : (H \times K)_x] = |H \times K| / |(H \times K)_x|.</math>
* ([[Cauchy–Frobenius lemma]]) Let {{math|''G''<sup>(''h'', ''k'')</sup>}} denote the elements fixed by the action of {{math|(''h'', ''k'')}}.  Then
*:<math>|H \,\backslash G / K| = \frac{1}{|H||K|}\sum_{(h, k) \in H \times K} |G^{(h, k)}|.</math>
* In particular, if {{math|''G''}}, {{math|''H''}}, and {{math|''K''}} 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 {{math|''H''}} and {{math|''K''}} both act by right multiplication on {{math|''G''}}.  Then {{math|''G''}} acts by left multiplication on the product of coset spaces {{math|''G'' / ''H'' × ''G'' / ''K''}}.  The orbits of this action are in one-to-one correspondence with {{math|''H'' \ ''G'' / ''K''}}.  This correspondence identifies {{math|(''xH'', ''yK'')}} with the double coset {{math|''Hx''<sup>−1</sup>''yK''}}.  Briefly, this is because every {{math|''G''}}-orbit admits representatives of the form {{math|(''H'', ''xK'')}}, and the representative {{math|''x''}} is determined only up to left multiplication by an element of {{math|''H''}}.  Similarly, {{math|''G''}} acts by right multiplication on {{math|''H'' \ ''G'' × ''K'' \ ''G''}}, and the orbits of this action are in one-to-one correspondence with the double cosets {{math|''H'' \ ''G'' / ''K''}}.  Conceptually, this identifies the double coset space {{math|''H'' \ ''G'' / ''K''}} with the space of relative configurations of an {{math|''H''}}-coset and a {{math|''K''}}-coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups {{math|''H''<sub>1</sub>, ..., ''H''<sub>''n''</sub>}}, the space of '''{{math|(''H''<sub>1</sub>, ..., ''H''<sub>''n''</sub>)}}-multicosets''' is the set of {{math|''G''}}-orbits of {{math|''G'' / ''H''<sub>1</sub> × ... × ''G'' / ''H''<sub>''n''</sub>}}.

The analog of [[Lagrange's theorem (group theory)|Lagrange's theorem]] for double cosets is false. This means that the size of a double coset need not [[divisor|divide]] the order of {{math|''G''}}.  For example, let {{math|1=''G'' = ''S''<sub>3</sub>}} be the [[symmetric group]] on three letters, and let {{math|''H''}} and {{math|''K''}} be the cyclic subgroups generated by the [[Transposition (mathematics)|transpositions]] {{math|(1 2)}} and {{math|(1 3)}}, respectively.  If {{math|''e''}} denotes the identity permutation, then
:<math>HeK = HK = \{ e, (1 2), (1 3), (1 3 2) \}.</math>
This has four elements, and four does not divide six, the order of {{math|''S''<sub>3</sub>}}.  It is also false that different double cosets have the same size.  Continuing the same example,
:<math>H(2 3)K = \{ (2 3), (1 2 3) \},</math>
which has two elements, not four.

However, suppose that {{math|''H''}} is normal.  As noted earlier, in this case the double coset space equals the left coset space {{math|''G'' / ''HK''}}.  Similarly, if {{math|''K''}} is normal, then {{math|''H'' \ ''G'' / ''K''}} is the right coset space {{math|''HK'' \ ''G''}}.  Standard results about left and right coset spaces then imply the following facts.

* {{math|1=<nowiki>|</nowiki>''HxK''<nowiki>|</nowiki> = <nowiki>|</nowiki>''HK''<nowiki>|</nowiki>}} for all {{math|''x''}} in {{math|''G''}}.  That is, all double cosets have the same cardinality.
* If {{math|''G''}} is finite, then {{math|1=<nowiki>|</nowiki>''G''<nowiki>|</nowiki> = <nowiki>|</nowiki>''HK''<nowiki>|</nowiki> ⋅ <nowiki>|</nowiki>''H'' \ ''G'' / ''K''<nowiki>|</nowiki>}}.  In particular, {{math|<nowiki>|</nowiki>''HK''<nowiki>|</nowiki>}} and {{math|<nowiki>|</nowiki>''H'' \ ''G'' / ''K''<nowiki>|</nowiki>}} divide {{math|<nowiki>|</nowiki>''G''<nowiki>|</nowiki>}}.

== Examples ==

* Let {{math|1=''G'' = ''S''<sub>''n''</sub>}} be the symmetric group, considered as [[permutation]]s of the set {{math|{1, ..., ''n''}}}.  Consider the subgroup {{math|1=''H'' = ''S''<sub>''n''−1</sub>}} which stabilizes {{math|''n''}}.  Then {{math|''S''<sub>''n''−1</sub> \ ''S''<sub>''n''</sub> / ''S''<sub>''n''−1</sub>}} consists of two double cosets.  One of these is {{math|1=''H'' = ''S''<sub>''n''−1</sub>}}, and the other is {{math|''S''<sub>''n''−1</sub> &gamma; ''S''<sub>''n''−1</sub>}} for any permutation {{math|&gamma;}} which does not fix {{math|''n''}}. This is contrasted with {{math|''S''<sub>''n''</sub> / ''S''<sub>''n''−1</sub>}}, 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 {{math|''G''}} be the group {{math|GL<sub>''n''</sub>('''R''')}}, and let {{math|''B''}} be the subgroup of [[upper triangular matrix|upper triangular matrices]].  The double coset space {{math|''B'' \ ''G'' / ''B''}} is the [[Bruhat decomposition]] of {{math|''G''}}.  The double cosets are exactly {{math|''BwB''}}, where {{math|''w''}} ranges over all [[permutation matrix|n-by-n permutation matrices]]. For instance, if {{math|1=''n'' = 2}}, 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 cosets ==

Suppose that {{math|''G''}} is a group and that {{math|''H''}}, {{math|''K''}}, and {{math|''L''}} are subgroups.  Under certain finiteness conditions, there is a product on the [[free abelian group]] generated by the {{math|(''H'', ''K'')}}- and {{math|(''K'', ''L'')}}-double cosets with values in the free abelian group generated by the {{math|(''H'', ''L'')}}-double cosets.  This means there is a [[Bilinear_map#Modules|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 {{math|''G''}} is finite.  To define the product, reinterpret these free abelian groups in terms of the [[group ring|group algebra]] of {{math|''G''}} as follows.  Every element of {{math|'''Z'''[''H'' \ ''G'' / ''K'']}} has the form
:<math>\sum_{HxK \in H \backslash G / K} f_{HxK} \cdot [HxK],</math>
where {{math|<nowiki>{</nowiki> ''f''<sub>''HxK''</sub> <nowiki>}</nowiki>}} is a set of [[integer]]s indexed by the elements of {{math|''H'' \ ''G'' / ''K''}}.  This element may be interpreted as a {{math|'''Z'''}}-valued function on {{math|''H'' \ ''G'' / ''K''}}, specifically, {{math|''HxK'' ↦ ''f''<sub>''HxK''</sub>}}.  This function may be pulled back along the projection {{math|''G'' → ''H'' \ ''G'' / ''K''}} which sends {{math|''x''}} to the double coset {{math|''HxK''}}.  This results in a function {{math|''x'' ↦ ''f''<sub>''HxK''</sub>}}.  By the way in which this function was constructed, it is left invariant under {{math|''H''}} and right invariant under {{math|''K''}}.  The corresponding element of the group algebra {{math|'''Z'''[''G'']}} is
:<math>\sum_{x \in G} f_{HxK} \cdot [x],</math>
and this element is invariant under left multiplication by {{math|''H''}} and right multiplication by {{math|''K''}}.  Conceptually, this element is obtained by replacing {{math|''HxK''}} by the elements it contains, and the finiteness of {{math|''G''}} ensures that the sum is still finite.  Conversely, every element of {{math|'''Z'''[''G'']}} which is left invariant under {{math|''H''}} and right invariant under {{math|''K''}} is the pullback of a function on {{math|'''Z'''[''H'' \ ''G'' / ''K'']}}.  Parallel statements are true for {{math|'''Z'''[''K'' \ ''G'' / ''L'']}} and {{math|'''Z'''[''H'' \ ''G'' / ''L'']}}.

When elements of {{math|'''Z'''[''H'' \ ''G'' / ''K'']}}, {{math|'''Z'''[''K'' \ ''G'' / ''L'']}}, and {{math|'''Z'''[''H'' \ ''G'' / ''L'']}} are interpreted as invariant elements of {{math|'''Z'''[''G'']}}, then the product whose existence was asserted above is precisely the multiplication in {{math|'''Z'''[''G'']}}.  Indeed, it is trivial to check that the product of a left-{{math|''H''}}-invariant element and a right-{{math|''L''}}-invariant element continues to be left-{{math|''H''}}-invariant and right-{{math|''L''}}-invariant.  The bilinearity of the product follows immediately from the bilinearity of multiplication in {{math|'''Z'''[''G'']}}.  It also follows that if {{math|''M''}} is a fourth subgroup of {{math|''G''}}, then the product of {{math|(''H'', ''K'')}}-, {{math|(''K'', ''L'')}}-, and {{math|(''L'', ''M'')}}-double cosets is associative.  Because the product in {{math|'''Z'''[''G'']}} corresponds to convolution of functions on {{math|''G''}}, this product is sometimes called the convolution product.

An important special case is when {{math|1=''H'' = ''K'' = ''L''}}.  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 {{math|'''Z'''[''H'' \ ''G'' / ''H'']}} into an [[ring (mathematics)|associative ring]] whose identity element is the class of the trivial double coset {{math|[''H'']}}.  In general, this ring is [[non-commutative ring|non-commutative]].  For example, if {{math|1=''H'' = <nowiki>{</nowiki>1<nowiki>}</nowiki>}}, then the ring is the group algebra {{math|'''Z'''[''G'']}}, and a group algebra is a [[commutative ring]] if and only if the underlying group is [[abelian group|abelian]].

If {{math|''H''}} is normal, so that the {{math|''H''}}-double cosets are the same as the elements of the quotient group {{math|''G'' / ''H''}}, then the product on {{math|'''Z'''[''H'' \ ''G'' / ''H'']}} is the product in the group algebra {{math|'''Z'''[''G'' / ''H'']}}.  In particular, it is the usual convolution of functions on {{math|''G'' / ''H''}}.  In this case, the ring is commutative if and only if {{math|''G'' / ''H''}} is abelian, or equivalently, if and only if {{math|''H''}} contains the [[commutator subgroup]] of {{math|''G''}}.

If {{math|''H''}} is not normal, then {{math|'''Z'''[''H'' \ ''G'' / ''H'']}} may be commutative even if {{math|''G''}} is [[non-abelian group|non-abelian]].  A classical example is the product of two [[Hecke operator]]s.  This is the product in the Hecke algebra, which is commutative even though the group {{math|''G''}} 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 pair]]s.

When the group {{math|''G''}} 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 {{math|'''Z'''[''G'']}} is replaced by an algebra of functions such as {{math|''L''<sup>2</sup>(''G'')}} or {{math|''C''<sup>∞</sup>(''G'')}}, and the sums are replaced by [[integral]]s.  The product still corresponds to convolution.  For instance, this happens for the [[Hecke algebra of a locally compact group]].

== Applications ==

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 group]]s for more information about this action.

Double cosets are important in connection with [[representation theory]], when a representation of {{math|''H''}} is used to construct an [[induced representation]] of {{math|''G''}}, which is then [[restricted representation|restricted]] to {{math|''K''}}.  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 {{math|''K''}} can form a [[commutative ring]] under [[convolution]]: see [[Gelfand pair]].

In geometry, a [[Clifford–Klein form]] is a double coset space {{math|Γ\''G''/''H''}}, where {{math|''G''}} is a [[reductive Lie group]], {{math|''H''}} is a closed subgroup, and {{math|Γ}} is a discrete subgroup (of {{math|''G''}}) that acts [[properly discontinuously]] on the [[homogeneous space]] {{math|''G''/''H''}}.

In [[number theory]], the [[Hecke operator|Hecke algebra]] corresponding to a [[congruence subgroup]] {{math|''Γ''}} 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 {{math|''m''}} and {{math|''N''}} 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 {{math|''a'', ''b'', ''c''}} does not affect the answer).

==References==
{{reflist}}

[[Category:Group theory]]