nLab
double coset

Contents

Idea

Given a group GG and two subgroups HH and KK, then the set of double cosets, H\G/KH \backslash G/K, is composed of equivalence classes of elements of GG, where two elements are regarded as equivalent if they differ by left multiplication with an element in HH and right multiplication with an element of KK.

This construction generalises to topological groups and Lie groups, where it is called a double coset space. The double coset space for GG a compact Lie group and HH and KK closed connected subgroups, with HH acting freely on G/KG/K, is also known as a biquotient (but note that this term is also used, e.g., here, in a more general situation for the orbit space where HH is a closed subgroup of G×GG \times G, and acts via (h 1,h 2)g=h 1gh 2 1(h_1, h_2) g = h_1 g h_2^{-1}).

From the nPOV, a weak form of the double coset construction is often more natural, H\\G//KH \backslash \backslash G//K, and can be defined as the homotopy pullback of maps of delooped groups,BHBG\mathbf{B} H \to \mathbf{B} G and BKBG\mathbf{B} K \to \mathbf{B} G. This holds more generally for any three ∞-groups, with any homomorphisms HGH \to G and KGK \to G.

Properties

Mackey’s formula

Double coset decompositions are useful in representation theory, for example in George Mackey‘s formula for the restriction back to HH of a module induced from KK. Let WW be a representation of KK. Then

Res H GInd K GW[g]H\G/KInd K g HW g, Res^G_H Ind^G_K W \cong \underset{[g] \in H \backslash G/K} \bigoplus Ind^H_{K_g} W_g \,,

where K gHgKg 1K_g \coloneqq H \intersection g K g^{-1}, and W gW_g is the representation of K gK_g which is WW as a vector space, but with action xw=(g 1xg)wx \cdot w = (g^{-1} x g) w where now g 1xgKg^{-1} x g \in K. This formula, sometimes called Mackey’s decomposition theorem, amounts to the Beck-Chevalley condition for the homotopy pullback square. It appears, as the Mackey axiom, in one of the original definitions of Mackey functor (Bouc 02).

References

  • Serge Bouc, Mackey functors, (pdf)

Last revised on August 7, 2017 at 04:19:18. See the history of this page for a list of all contributions to it.