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 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.

An important case is when H = K, when there is a kind of product:

HyH·HyH

is a union of double cosets. In some contexts, for example for finite groups, this can be made the basis for an associated ring.

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.