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

H

and

k

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 $[K : K \cap g -1 Hg]$ and the number of left cosets of $K$ in $HgK$ is the index $[g -1 Hg : K \cap g -1 Hg]$ .^[1]

Double cosets are in fact orbits for the group action of $H \times K$ on $G$ with $(h, k)\cdotg := hgk -1$ . 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 $Hg_{1}K$ and $Kg_{2}L$ , we decompose them respectively into right and left cosets $\textstyle Hg_{1}K=\coprod _{i}Ha_{i}$ and $\textstyle Kg_{2}L=\coprod _{j}b_{j}L$ . If we write $c_{g}=\left|\{(i,j):a_{i}b_{j}\in Hg\}\right|$ , then we can define the product of $Hg_{1}K$ with $Kg_{2}L$ as the formal sum $\textstyle Hg_{1}K\cdot Kg_{2}L=\sum _{g\in H\setminus G}c_{g}HgL$ .

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

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