Double coset

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 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 space of double cosets can be written

H\G/K.

Algebraic structure

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

Given two double cosets Hg₁K and Kg₂L, we decompose each into right cosets $H g_1 K = \coprod_i H a_i$ and $K g_2 L = \coprod_j K b_j$. If we write $c_g = \left | \{ (i,j) : a_i b_j \in H g \} \right |$, then we can define the product of Hg₁K with Kg₂L as the formal sum

$H g_1 K \cdot K g_2 L = \sum_{g \in H \backslash G} c_g H g L.$

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 $\Gamma \backslash \mathrm{GL}_2^+(\mathbb{Q}) / \Gamma$; 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 Γ₀(N)gΓ₀(N) or Γ₁(N)gΓ₁(N), where $g= \begin{pmatrix} 1 & 0 \\ 0 & m \end{pmatrix}$ (these have different properties depending on whether m and N are coprime or not), and the diamond operators $\langle d \rangle$ given by the double cosets $\Gamma_1(N) \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Gamma_1(N)$ where $d \in (\mathbb{Z}/N\mathbb{Z})^\times$ and we require $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$ (the choice of a, b, c does not affect the answer).