Editing Double coset (section)

== 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).