Double coset

In mathematics, given a group G and two subgroups H and K, not necessarily distinct, a double coset (or more precisely an (H,K) double coset) is a set HgK for some fixed element g in G. Equivalently, an (H,K) double coset in 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.

The basic properties of double cosets follow immediately from the fact that they are equivalence classes; namely, two double cosets HxK and HyK are either disjoint or identical^[1] and G is partitioned into its (H,K) double cosets.^[2]

Furthermore, each double coset HgK is a union of ordinary right cosets Hy of H in G and left cosets zK of K in G. In particular, the number of right cosets of H in HgK is the index [K : K ∩ g⁻¹Hg] and the number of left cosets of K in HgK is the index [g⁻¹Hg : K ∩ g⁻¹Hg].^[1]

Double cosets are in fact orbits for the group action of H×K on G with (h, k)⋅g := hgk⁻¹. The set of double cosets is often denoted by

$H \backslash G / K.$

Algebraic structure

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

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

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 $\Gamma_0(N) g \Gamma_0(N)$ or $\Gamma_1(N) g \Gamma_1(N)$, where $g= \left( \begin{smallmatrix} 1 & 0 \\ 0 & m \end{smallmatrix} \right)$ (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 $\Gamma_1(N) \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \Gamma_1(N)$ where $d \in (\mathbb{Z}/N\mathbb{Z})^\times$ and we require $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)\in \Gamma_0(N)$ (the choice of a, b, c does not affect the answer).

References

1. Hall, Jr., Marshall (1959), The Theory of Groups, New York: Macmillan, pp. 14 – 15
2. Bechtell, Homer (1971), The Theory of Groups, Addison-Wesley, p. 101