Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 2nd 2017
   • (edited Aug 2nd 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexThere are certainly plenty of important examples as appear for instance in number theory: Hecke algebras naturally arise from them, given assumptions of pushforwards.

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

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 2nd 2017
   • (edited Aug 2nd 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThe 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](https://en.wikipedia.org/wiki/Clifford%E2%80%93Klein_form). I wonder what are the 'sensible' restrictions on the quotienting groups. I see from [here](https://arxiv.org/abs/math/0409559): >If $G_0 \subset G$ is a closed subgroup of a Lie group and $\Gamma \subset G$ is a discrete subgroup, call the (possibly singular) space $\Gamma \backslash G/G_0$ a double coset space. If $\Gamma \subset G$ acts on the left on $G/G_0$ freely and properly, call the smooth manifold $\Gamma\backslash G/G_0$ a locally Klein geometry.

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

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo people also allow both groups to be topological. E.g., at the top of p.320 [here](https://books.google.fr/books?id=t-aM8bXHvy4C&pg=PA320) there is the construction where $G/K$ is $S^7$ and $H \backslash G /K$ is $S^4$. 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](https://nforum.ncatlab.org/discussion/7254/quest-for-generalized-cohomology-theory-for-mbrane-charges/?Focus=64607#Comment_64607).]

   So people also allow both groups to be topological. E.g., at the top of p.320 here there is the construction where $G/K$ is $S^7$ and $H \backslash G /K$ is $S^4$. 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.]

5. • CommentRowNumber5.
   • CommentAuthorPeter Heinig
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItexIncidentally, 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 $G$, not necessarily finite, there exists a cardinal $\kappa$ such that the [lexicographic product](https://en.wikipedia.org/wiki/Lexicographic_product_of_graphs) of $G$ with the coclique on $\kappa$ ($=$ free graph on $\kappa$) 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](https://en.wikipedia.org/wiki/Schreier_coset_graph) 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](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002474573&physid=PHYS_0445). 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*.

   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 $G$, not necessarily finite, there exists a cardinal $\kappa$ such that the lexicographic product of $G$ with the coclique on $\kappa$ ($=$ free graph on $\kappa$) 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.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOk, 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexJohn B has some stuff in episode 4 of [this](http://math.ucr.edu/home/baez/groupoidification/) : >A SPAN OF GROUPOIDS EQUIPPED WITH CERTAIN EXTRA STUFF IS THE SAME AS A DOUBLE COSET.

   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.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexDoes that construction at [[Hecke category]] work in general? So given subgroups $H$ and $K$ of $G$, form the pullback of $B H \to B G$ and $B K \to B G$.

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

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeAug 2nd 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. This is spelled out as an example in the entry _[[homotopy limit]]_ ([here](https://ncatlab.org/nlab/show/homotopy%20limit#DeloopedSubgroupOverGroup)). (I seem to remember that this is from the very early days of the nLab.)

   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.)

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 2nd 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexSo that then works for $H$ and $K$ different, presumably. I guess for $\infty$-groups, as usual we just have any morphisms to $G$. Presumably people writing $H \backslash G /K$ generally mean the strict double quotient.

    So that then works for $H$ and $K$ different, presumably. I guess for $\infty$-groups, as usual we just have any morphisms to $G$.

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

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 2nd 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes!

    Yes!

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI 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 $B H \to B G$ and $B K \to B G$, it all seems quite [pull push like](https://ncatlab.org/nlab/show/integral+transforms+on+sheaves) in the context $B G$.

    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 $B H \to B G$ and $B K \to B G$, it all seems quite pull push like in the context $B G$.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2017
    • (edited Aug 4th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexTrue, that [formula](https://ncatlab.org/nlab/show/double+coset#MackeyFormula) looks like it wants to express the Beck-Chevalley condition for pull-push through $$ \array{ && K \backslash G / H \\ & \swarrow && \searrow \\ B K && (pb) && B H \\ & \searrow && \swarrow \\ && B G } $$ By the way, I expanded the subscript under the direct sum to $[g] \in H \backslash G / K$. I hope that's correct.

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

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

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

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexYes, the notes I was looking at had $g$ as representative of $H g K$.

    Yes, the notes I was looking at had $g$ as representative of $H g K$.

15. • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexTom Dieck is doing something similar on p. 164 [here](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-induction.pdf) except around $$ \array{ && G/H \times G/K \\ & \swarrow && \searrow \\ G/H && (pb) && G/K \\ & \searrow && \swarrow \\ && G/G } $$ >This is the so called double coset formula which one can never remember and which is avoided by this axiomatic treatment.

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

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

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

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThe worrying thing is how slowly one learns. There I was [nearly ten years ago](https://golem.ph.utexas.edu/category/2007/10/concrete_groups_and_axiomatic.html#c012923) asking about a Beck-Chevalley condition for groups. Simon Willerton suggested something more-or-less like Mackey restriction, then John Baez [points out](https://golem.ph.utexas.edu/category/2007/10/concrete_groups_and_axiomatic.html#c012929): >The key technical tool in Jeffrey Morton’s [thesis](https://arxiv.org/pdf/0710.0032.pdf) 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.

    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.