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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 19th 2015
   • (edited Aug 19th 2015)
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   Author: Urs
   Format: MarkdownItexhad added to _[[finite group]]_ two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres. What I'd really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.

   had added to finite group two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres.

   What I’d really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 19th 2015
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   Author: David_Corfield
   Format: MarkdownItexIs this in a similar context to >In the case when the microscopic gauge group is Higgsed down to a finite group $G$, a complete classification (at least if no further global symmetries are postulated) has been given by Dijkgraaf and Witten [1]. Namely, topological actions in d dimensions are classified by degree-d cohomology classes for G with coefficients in U(1). ([Higher symmetry and gapped phases of gauge theories](http://www.cims.nyu.edu/~tschinke/books/maxim14/submitted/HigherSymmetry.pdf)] [1] R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” Commun. Math. Phys. 129, 393 (1990).

   Is this in a similar context to

   In the case when the microscopic gauge group is Higgsed down to a finite group $G$, a complete classification (at least if no further global symmetries are postulated) has been given by Dijkgraaf and Witten [1]. Namely, topological actions in d dimensions are classified by degree-d cohomology classes for G with coefficients in U(1). (Higher symmetry and gapped phases of gauge theories]

   [1] R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” Commun. Math. Phys. 129, 393 (1990).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 19th 2015
   • (edited Aug 19th 2015)
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   Author: Urs
   Format: MarkdownItexYes. I am still trying, from the [other thread](http://nforum.ncatlab.org/discussion/6694/from-higher-to-exceptional-geometry/?Focus=54171#Comment_54171), to see if there is a way to find the [[enhanced gauge symmetry]] of M-theory on ADE ordifolds directly, without recourse to heterotic duality. Now the extended action functional of the M2-brane is a 2-gerbe connection $\mathbf{L} : X \longrightarrow \mathbf{B}^3 (\mathbb{C}/\Gamma)_{conn}$ and so as we do the higher geometric prequantization of the M2 then the phase space over a 2-surface $\Sigma_2$ is the mapping stack $[\Pi(\Sigma_2), X]$. This is just $X$ itself at all smooth points, reflecting the fact that the critical M2-configurations there are just shrunken to single points. But at each orbifold fixed point in $X$ of the form $\ast / G \simeq \mathbf{B}G$ this phase space becomes that of $G$-Chern-Simons theory on $\Sigma_2$ and specifically of $G$-Dijkgraaf-Witten theory since $G$ is finite. Hence there is "gauge enhancement" at these points in a sense, and I am trying to see if this is the right kind of sense. Now $\mathbf{L}$ is constrained to be a [[definite globalization of WZW term|definite globalization]] hence its form at these orbifold singularities is not completely arbitrary. One gets that it is of the form $\ast /G \simeq \mathbf{B}G \to \flat \mathbf{B}^3 (\mathbb{C}/\mathbb{Z}) \to \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn}$. In other words, it is given there by a degree-3 group cocycle on $G$ with coefficients in $\mathbb{C}/\mathbb{Z}$ (a Dijkgraaf-Witten action functional, if you wish, as in the quote that you give). So what's the cohomology in degree-3 with coefficients in $\mathbb{C}/\mathbb{Z}$ (or $\mathbb{R}/\mathbb{Z}$) of the finite groups appearing in the [[ADE -- table]], the finite subgroups of $SO(3)$? I see that the icosahedral group has group cohomology with _integral_ coefficients concentrated in degrees $4k$. What exactly is it in degree $4$? Is it $\mathbb{Z}$ there?

   Yes. I am still trying, from the other thread, to see if there is a way to find the enhanced gauge symmetry of M-theory on ADE ordifolds directly, without recourse to heterotic duality. Now the extended action functional of the M2-brane is a 2-gerbe connection $\mathbf{L} : X \longrightarrow \mathbf{B}^3 (\mathbb{C}/\Gamma)_{conn}$ and so as we do the higher geometric prequantization of the M2 then the phase space over a 2-surface $\Sigma_2$ is the mapping stack $[\Pi(\Sigma_2), X]$. This is just $X$ itself at all smooth points, reflecting the fact that the critical M2-configurations there are just shrunken to single points. But at each orbifold fixed point in $X$ of the form $\ast / G \simeq \mathbf{B}G$ this phase space becomes that of $G$-Chern-Simons theory on $\Sigma_2$ and specifically of $G$-Dijkgraaf-Witten theory since $G$ is finite. Hence there is “gauge enhancement” at these points in a sense, and I am trying to see if this is the right kind of sense.

   Now $\mathbf{L}$ is constrained to be a definite globalization hence its form at these orbifold singularities is not completely arbitrary. One gets that it is of the form $\ast /G \simeq \mathbf{B}G \to \flat \mathbf{B}^3 (\mathbb{C}/\mathbb{Z}) \to \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn}$. In other words, it is given there by a degree-3 group cocycle on $G$ with coefficients in $\mathbb{C}/\mathbb{Z}$ (a Dijkgraaf-Witten action functional, if you wish, as in the quote that you give).

   So what’s the cohomology in degree-3 with coefficients in $\mathbb{C}/\mathbb{Z}$ (or $\mathbb{R}/\mathbb{Z}$) of the finite groups appearing in the ADE – table, the finite subgroups of $SO(3)$?

   I see that the icosahedral group has group cohomology with integral coefficients concentrated in degrees $4k$. What exactly is it in degree $4$? Is it $\mathbb{Z}$ there?

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 19th 2015
   • (edited Aug 19th 2015)
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   Author: David_Corfield
   Format: MarkdownItexSo the cyclic groups are apparently [known](http://mathoverflow.net/q/144603/447) >degree-3 cohomology of Z/n with coefficients in R/Z (with a trivial action of Z/n) is isomorphic to Z/n. There's cohomology of dihedral D8 for abelian coefficients [here](http://groupprops.subwiki.org/wiki/Group_cohomology_of_dihedral_group:D8) The same site only has integral homology for the tetrahedral group [here](http://groupprops.subwiki.org/wiki/Group_cohomology_of_alternating_group:A4). Same for $S_4$ (octohedral) [here](http://groupprops.subwiki.org/wiki/Group_cohomology_of_symmetric_group:S4) and [SL(2, 5)](http://groupprops.subwiki.org/wiki/Group_cohomology_of_special_linear_group:SL(2,5)). Edit: should be $A_5$ not $SL(2, 5)$.

   So the cyclic groups are apparently known

   degree-3 cohomology of Z/n with coefficients in R/Z (with a trivial action of Z/n) is isomorphic to Z/n.

   There’s cohomology of dihedral D8 for abelian coefficients here

   The same site only has integral homology for the tetrahedral group here. Same for $S_4$ (octohedral) here and SL(2, 5).

   Edit: should be $A_5$ not $SL(2, 5)$.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 19th 2015
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   Author: David_Corfield
   Format: MarkdownItexOh, doesn't $H^k(B G, R/Z) = H^{k+1}(B G, Z)$, because of the long exact sequence from $0 \to Z \to R \to R/Z \to 0$, and cohomology with coefficients in $R$ being trivial?

   Oh, doesn’t $H^k(B G, R/Z) = H^{k+1}(B G, Z)$, because of the long exact sequence from $0 \to Z \to R \to R/Z \to 0$, and cohomology with coefficients in $R$ being trivial?

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 19th 2015
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   Author: David_Corfield
   Format: MarkdownItexIntegral cohomology of $S_4$ is on p. 2 [here](http://arxiv.org/pdf/1309.4238.pdf).

   Integral cohomology of $S_4$ is on p. 2 here.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 19th 2015
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   Author: Urs
   Format: MarkdownItexThanks!! I have added these pointers to _[[dihedral group]]_ and _[[tetrahedral group]]_ and maybe elsewhere. Have to run now. Thanks again.

   Thanks!!

   I have added these pointers to dihedral group and tetrahedral group and maybe elsewhere.

   Have to run now. Thanks again.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 19th 2015
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   Author: David_Corfield
   Format: MarkdownItexWait a minute. You're only considering orientation preserving transformations, as subgroups of $SO(3)$. So you should have tetrahedral group as $A_4$, octohedral group as $S_4$, and icosohedral group as $A_5$. Are these official names? I see [Mathworld](http://mathworld.wolfram.com/TetrahedralGroup.html) has 'tetrahedral group' down as the full symmetry group with inversions, so $S_4$.

   Wait a minute. You’re only considering orientation preserving transformations, as subgroups of $SO(3)$. So you should have tetrahedral group as $A_4$, octohedral group as $S_4$, and icosohedral group as $A_5$.

   Are these official names? I see Mathworld has ’tetrahedral group’ down as the full symmetry group with inversions, so $S_4$.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 19th 2015
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   Author: DavidRoberts
   Format: MarkdownItexThe 2-group extensions of Platonic groups by BU(1) were done by a student of Nora Ganter -- I think it was Narthana Epa. Sorry for no more, I'm off to bed

   The 2-group extensions of Platonic groups by BU(1) were done by a student of Nora Ganter – I think it was Narthana Epa. Sorry for no more, I’m off to bed

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015
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    Author: David_Corfield
    Format: MarkdownItexRe #9, that's in the paper [Platonic 2-groups](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf).

    Re #9, that’s in the paper Platonic 2-groups.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)
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    Author: Urs
    Format: MarkdownItex@David C., I have added the distinctions to _[[tetrahedral group]]_ and _[[octahedral group]]_ and _[[symmetric group]]_. @David R. thanks! I have found it: * [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

    @David C., I have added the distinctions to tetrahedral group and octahedral group and symmetric group.

    @David R. thanks! I have found it:

    • Narthana Epa, Platonic 2-groups, 2010 (pdf)
12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020
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    Author: Urs
    Format: MarkdownItexadded this pointer: * [[Louis Solomon]], _The representation of finite groups in algebraic number fields_, J. Math. Soc. Japan Volume 13, Number 2 (1961), 144-164 ([euclid:euclid.jmsj/1261147137](https://projecteuclid.org/euclid.jmsj/1261147137)) <a href="https://ncatlab.org/nlab/revision/diff/finite+group/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/finite+group/23">v23</a>, <a href="https://ncatlab.org/nlab/show/finite+group">current</a>

    added this pointer:

    • Louis Solomon, The representation of finite groups in algebraic number fields, J. Math. Soc. Japan Volume 13, Number 2 (1961), 144-164 (euclid:euclid.jmsj/1261147137)

    diff, v23, current

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2021
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    Author: Urs
    Format: MarkdownItexadded pointer to: * G. A. Miller, H. F. Blichfeldt, [[Leonard E. Dickson]], _Theory and applications of finite groups_, Dover, New York, 1916 ([ark:/13960/t0ht2kb4x](https://archive.org/details/theoryapplicatio00milluoft/)) <a href="https://ncatlab.org/nlab/revision/diff/finite+group/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/finite+group/26">v26</a>, <a href="https://ncatlab.org/nlab/show/finite+group">current</a>

    added pointer to:

    • G. A. Miller, H. F. Blichfeldt, Leonard E. Dickson, Theory and applications of finite groups, Dover, New York, 1916 (ark:/13960/t0ht2kb4x)

    diff, v26, current

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022
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    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 7 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$ <a href="https://ncatlab.org/nlab/revision/diff/finite+group/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/finite+group/28">v28</a>, <a href="https://ncatlab.org/nlab/show/finite+group">current</a>

    added pointer to:

    • Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 7 of: Symmetry (2021) $[$pdf$]$

    diff, v28, current