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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017
    • (edited Aug 2nd 2017)

    I see we don’t have an entry for double coset (space). Where it appears, any link is to coset (space). Should we just have a section on that page, or does it merit its own page?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 2nd 2017

    There are certainly plenty of important examples as appear for instance in number theory: Hecke algebras naturally arise from them, given assumptions of pushforwards.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017
    • (edited Aug 2nd 2017)

    The five pages containing the term are: Gram-Schmidt process, Hecke algebra, automorphic form, geometric Satake equivalence, hypermonoid.

    Probably sounds like it merits its own page then. Examples include Clifford-Klein form. I wonder what are the ’sensible’ restrictions on the quotienting groups. I see from here:

    If G 0GG_0 \subset G is a closed subgroup of a Lie group and ΓG\Gamma \subset G is a discrete subgroup, call the (possibly singular) space Γ\G/G 0\Gamma \backslash G/G_0 a double coset space. If ΓG\Gamma \subset G acts on the left on G/G 0G/G_0 freely and properly, call the smooth manifold Γ\G/G 0\Gamma\backslash G/G_0 a locally Klein geometry.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017

    So people also allow both groups to be topological. E.g., at the top of p.320 here there is the construction where G/KG/K is S 7S^7 and H\G/KH \backslash G /K is S 4S^4. And you can make an exotic 7-sphere in similar fashion.

    [In case people are wondering why the sudden interest, I was thinking a little about Urs’s “Cartan geometry with singularities” here.]

    • CommentRowNumber5.
    • CommentAuthorPeter Heinig
    • CommentTimeAug 2nd 2017

    Incidentally, just to add one small example lending support to (cf. #3)

    Probably sounds like it merits its own page then.

    If I recall correctly, it is a famous theorem in algebraic graph theory, proved in the 1960s, that

    • for each vertex-transitive graph GG, not necessarily finite, there exists a cardinal κ\kappa such that the lexicographic product of GG with the coclique on κ\kappa (== free graph on κ\kappa) is a Cayley graph.

    Sloganesquely put:

    • Any vertex-transitive graph inflates to a Cayley-graph.

    Note that lexicographic product with free graphs is what in graph-theory is often called “blowing-up a graph”.

    And (which is why this is relevant to double-cosets), the first (and to my knowledge only) proof of this generalizes Schreier coset graphs to what one may call double coset graphs, and, roughly speaking, reasons like (vertex-transitive)\Rightarrow(has representation via double cosets)\Rightarrow(has representation as a Cayley graph).

    Here, certain double cosets, invariant under a inversion-map, are used to define a symmetric adjacency relation of the Cayley-graph-to-be-constructed.

    Reference: Theorem 4 herein.

    Remark. Sabidussi’s above-cited paper is a very often cited one; it also contains the theorem that—barring a few trivialities—any vertex-transitive graph has non-abelian automorphism group.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017

    Ok, so double coset is launched.

    Re my list in #3, I only searched for “double coset” on the internal search engine, so missed those of the form “double coset”. When I get a moment I’ll replace with links to ’double coset’.

    I started also local Klein geometry and Clifford-Klein form. They seem mighty similar constructions.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 2nd 2017

    John B has some stuff in episode 4 of this :

    A SPAN OF GROUPOIDS EQUIPPED WITH CERTAIN EXTRA STUFF IS THE SAME AS A DOUBLE COSET.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017

    Does that construction at Hecke category work in general? So given subgroups HH and KK of GG, form the pullback of BHBGB H \to B G and BKBGB K \to B G.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 2nd 2017

    Yes. This is spelled out as an example in the entry homotopy limit (here).

    (I seem to remember that this is from the very early days of the nLab.)

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017

    So that then works for HH and KK different, presumably. I guess for \infty-groups, as usual we just have any morphisms to GG.

    Presumably people writing H\G/KH \backslash G /K generally mean the strict double quotient.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 2nd 2017

    Yes!

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017

    I added in a few bits and pieces, including Mackey’s restriction formula for restricting induced representations. Is there not a clever abstract general way to compose dependent sum and base change? Given double cosets as the pullback of BHBGB H \to B G and BKBGB K \to B G, it all seems quite pull push like in the context BGB G.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2017
    • (edited Aug 4th 2017)

    True, that formula looks like it wants to express the Beck-Chevalley condition for pull-push through

    K\G/H BK (pb) BH BG \array{ && K \backslash G / H \\ & \swarrow && \searrow \\ B K && (pb) && B H \\ & \searrow && \swarrow \\ && B G }

    By the way, I expanded the subscript under the direct sum to [g]H\G/K[g] \in H \backslash G / K. I hope that’s correct.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017

    Yes, the notes I was looking at had gg as representative of HgKH g K.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017

    Tom Dieck is doing something similar on p. 164 here except around

    G/H×G/K G/H (pb) G/K G/G \array{ && G/H \times G/K \\ & \swarrow && \searrow \\ G/H && (pb) && G/K \\ & \searrow && \swarrow \\ && G/G }

    This is the so called double coset formula which one can never remember and which is avoided by this axiomatic treatment.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017

    The worrying thing is how slowly one learns. There I was nearly ten years ago asking about a Beck-Chevalley condition for groups. Simon Willerton suggested something more-or-less like Mackey restriction, then John Baez points out:

    The key technical tool in Jeffrey Morton’s thesis was a Beck–Chevalley condition applying to groupoids. You’ll see the Beck–Chevalley square in diagram (125) on page 69, and the result on following pages. It’s embedded in a specific technical context that’s fascinating to me, but maybe not to you… he’s constructing an extended topological quantum field theory.

    In some ways this Beck–Chevalley condition for groupoid representations subsumes your idea – but there’s an important subtlety, which I’ll discuss.

    Jeffrey showed that given a weak pullback of groupoids, some restriction of an induced representation is isomorphic to an induced representation of a restricted representation.

    This is the real subtlety. Jeffrey only got his result to work with weak pullbacks! These are clearly the “morally correct” thing to use with groupoids… but I seem to recall that even with groups, he needed to think of them as 1-object groupoids and use the weak pullback to get a Beck–Chevalley condition to hold. Even in this case, the weak pullback is different from the strict one!

    Later Todd pointed out the BC-condition holds for groupoids but not categories as base types.