# Double coset

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In mathematics, an (*H*,*K*) **double coset** in *G*, where *G* is a group and *H* and *K* are subgroups of *G*, is an equivalence class for the equivalence relation defined on *G* by

*x*~*y*if there are*h*in*H*and*k*in*K*with*hxk*=*y*.

Then each double coset is of the form *HxK*, and *G* is partitioned into its (*H*,*K*) double cosets; each of them is a union of ordinary right cosets *Hy* of *H* in *G* and left cosets *zK* of K in G. In another aspect, these are in fact orbits for the group action of *H*×*K* on *G* with *H* acting by left multiplication and *K* by inverse right multiplication. The set of double cosets can be written

*H*\*G*/*K*.

## Algebraic structure

It is possible to define a product operation of double cosets in an associated ring.

Given two double cosets and , we decompose each into right cosets and . If we write , then we can define the product of with as the formal sum .

An important case is when *H* = *K* = *L*, which allows us to define an algebra structure on the associated ring spanned by linear combinations of double cosets.

## Applications

Double cosets are important in connection with representation theory, when a representation of *H* is used to construct an induced representation of *G*, which is then restricted to *K*. The corresponding double coset structure carries information about how the resulting representation decomposes.

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

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

In number theory, the Hecke algebra corresponding to a congruence subgroup *Γ* of the modular group is spanned by elements of the double coset space ; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators corresponding to the double cosets or , where (these have different properties depending on whether *m* and *N* are coprime or not), and the diamond operators given by the double cosets where and we require (the choice of *a*, *b*, *c* does not affect the answer).