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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 29th 2019
   • (edited Aug 30th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere something elementary but curious, please set me straight if I am just missing the obvious: For $2 D_{2k}$ a [[binray dihedral group]], I want to count the number of *real* irrep summands in its *real* regular representation, hence I want this number $N_k$ of summands $$ \mathbb{R}[2 D_{2k}] \simeq \underset{ N_k \, \text{summands} }{ \underbrace{ \text{real irrep} \oplus \text{real irrep} \oplus \cdots \oplus \text{real irrep} }} $$ If we were over the complex numbers, the answer would of course be the sum of dimensions over distinct iso classes of irreps. So just to correct for the fact that we are now over the real numbers. I think the rule now is: _$N$ is the sum of real dimensions over real irrep classes, *except* for quaternionic type reps which instead contribute with half their dimension._ Right? Applying this counting to the first few examples, with dimensions read off from the first column of any character table, e.g. from the real character tables shown [here](https://www.maths.manchester.ac.uk/~jm/wiki/Representations/Dicyclic#Dic2), I get the following: | bin dihed. group $2 D_{2k}$ | number of real irreps in the real regular rep | $ = N_k$ | |--|--| | $\mathbb{Z}/2$ | $1 + 1$ | $= 2$ | | $\mathbb{Z}/4 = Dic_1$ | $1 + 1 + 2$ | $= 4$ | | $2 D_{4} = Dic_2$ | $1 + 1 + 1 + 1 + 4/2$ | $= 6$ | | $2 D_{6} = Dic_3$ | $1 + 1 + 2 + 2 + 4/2$ | $ = 8$ | | $2 D_{8} = Dic_4$ | $1 + 1 + 1 + 1 + 2 + 4/2 + 4/2$ | $ = 10$ | | $2 D_{10} = Dic_5$ | $1 + 1 + 2 + 2 + 2 + 4/2 + 4/2$ | $ = 12$ | | $2 D_{12} = Dic_6$ | $1 + 1 + 1 + 1 + 2 + 2 + 4/2 + 4/2 + 4/2$ | $ = 14$ | | $2 D_{14} = Dic_7$ | $1 + 1 + 2 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2$ | $ = 16$ | | $2 D_{16} = Dic_8$ | $1 + 1 + 1 + 1 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2 + 4/2$ | $ = 18$ | | $2 D_{18} = Dic_9$ | $1 + 1 + 2 + 2 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2 + 4/2 $ | $ = 20$ | | $2 D_{2k} = Dic_k$ | $\phantom{AAAA}$?? | $ = 2k + 2$ | Here the last line makes the evident guess for the general statement: $$ N_k \;=\; 2k + 2 \phantom{AA} \text{generally ??} $$ Is this right? In the above examples? In generality? What would be a general proof?

   Here something elementary but curious, please set me straight if I am just missing the obvious:

   For $2 D_{2k}$ a binray dihedral group, I want to count the number of real irrep summands in its real regular representation, hence I want this number $N_k$ of summands

   $$\mathbb{R}[2 D_{2k}] \simeq \underset{ N_k \, \text{summands} }{ \underbrace{ \text{real irrep} \oplus \text{real irrep} \oplus \cdots \oplus \text{real irrep} }}$$

   If we were over the complex numbers, the answer would of course be the sum of dimensions over distinct iso classes of irreps.

   So just to correct for the fact that we are now over the real numbers. I think the rule now is:

   $N$ is the sum of real dimensions over real irrep classes, except for quaternionic type reps which instead contribute with half their dimension.

   Right?

   Applying this counting to the first few examples, with dimensions read off from the first column of any character table, e.g. from the real character tables shown here, I get the following:

   bin dihed. group $2 D_{2k}$	number of real irreps in the real regular rep	$= N_k$
   $\mathbb{Z}/2$	$1 + 1$	$= 2$
   $\mathbb{Z}/4 = Dic_1$	$1 + 1 + 2$	$= 4$
   $2 D_{4} = Dic_2$	$1 + 1 + 1 + 1 + 4/2$	$= 6$
   $2 D_{6} = Dic_3$	$1 + 1 + 2 + 2 + 4/2$	$= 8$
   $2 D_{8} = Dic_4$	$1 + 1 + 1 + 1 + 2 + 4/2 + 4/2$	$= 10$
   $2 D_{10} = Dic_5$	$1 + 1 + 2 + 2 + 2 + 4/2 + 4/2$	$= 12$
   $2 D_{12} = Dic_6$	$1 + 1 + 1 + 1 + 2 + 2 + 4/2 + 4/2 + 4/2$	$= 14$
   $2 D_{14} = Dic_7$	$1 + 1 + 2 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2$	$= 16$
   $2 D_{16} = Dic_8$	$1 + 1 + 1 + 1 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2 + 4/2$	$= 18$
   $2 D_{18} = Dic_9$	$1 + 1 + 2 + 2 + 2 + 2 + 2 + 4/2 + 4/2 + 4/2 + 4/2$	$= 20$
   $2 D_{2k} = Dic_k$	$\phantom{AAAA}$??	$= 2k + 2$

   Here the last line makes the evident guess for the general statement:

   $$N_k \;=\; 2k + 2 \phantom{AA} \text{generally ??}$$

   Is this right? In the above examples? In generality? What would be a general proof?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 29th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexDon't think the $2D_{14}$ is right. It says 1 dim real reps alternate between four and two. Presumbably it's $1+1+2+2+2+2+4/2+4/2+4/2$. Seems an easy pattern for odd then even, but I don't know why.

   Don’t think the $2D_{14}$ is right. It says 1 dim real reps alternate between four and two. Presumbably it’s

   $1+1+2+2+2+2+4/2+4/2+4/2$.

   Seems an easy pattern for odd then even, but I don’t know why.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 29th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex$Dic_{2n}: 4 \times 1 + (n-1) \times 2 + n \times 4/2$ and $Dic_{2n +1}: 2 \times 1 + (n+1) \times 2 + n \times 4/2$.

   $Dic_{2n}: 4 \times 1 + (n-1) \times 2 + n \times 4/2$ and $Dic_{2n +1}: 2 \times 1 + (n+1) \times 2 + n \times 4/2$.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 29th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexYou can see this from [here](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups).

   You can see this from here.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Don't think the $2D_{14}$ is right. Thanks for catching this. Have fixed it now. (Took this from [groupnames Dic7](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Dic7.html) and forgot to add up $\rho_3$ with $\rho_4$ to make them real.)

   Don’t think the $2D_{14}$ is right.

   Thanks for catching this. Have fixed it now. (Took this from groupnames Dic7 and forgot to add up $\rho_3$ with $\rho_4$ to make them real.)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2019
   • (edited Aug 30th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> $Dic_{2n}: 4 \times 1 + (n-1) \times 2 + n \times 4/2$ and $Dic_{2n +1}: 2 \times 1 + (n+1) \times 2 + n \times 4/2$. > You can see this from [here](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups). Thanks! I find that page at groupprops hard to read, but I suppose I see it now. Okay, so then that's the answer to this maths question. Thanks! Now I am left with seeing if I understand the physics meaning behind this. Still a bit puzzled about that...

   $Dic_{2n}: 4 \times 1 + (n-1) \times 2 + n \times 4/2$ and $Dic_{2n +1}: 2 \times 1 + (n+1) \times 2 + n \times 4/2$.

   You can see this from here.

   Thanks! I find that page at groupprops hard to read, but I suppose I see it now.

   Okay, so then that’s the answer to this maths question. Thanks!

   Now I am left with seeing if I understand the physics meaning behind this. Still a bit puzzled about that…

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2019
   • (edited Aug 30th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy the way, for properly completing the pattern in low degrees, I feel that one should declare the following degenerate cases of binary dihedral / dicyclic groups $$ 2 D_{2} \coloneqq Dic_1 = \mathbb{Z}_4 $$ and $$ 2 D_{0} \coloneqq Dic_0 \coloneqq \mathbb{Z}_2 $$ (I have added these lines to [#1](#Comment_79857) now.) Being careful about these degenerate cases seems to be necessary in order to disentangle some subtleties in the string literature. For instance there is originally the idea that one can have a toroidal orientifold of the form $$ \mathbb{T}^4/\mathbb{Z}_N $$ for any even $N$, with the reflection being the element $[N/2] \in \mathbb{Z}_N$ ([Gimon-Johnson 96, p. 6-7 (7-8 of 32)](https://ncatlab.org/nlab/show/orientifold#GimonJohnson96)). But then people find that actually in the case $N = 6$ this leads to inconsistency and they restrict attention to just $N = 2$ and $N = 4$ ([Buchel-Shiu-Tye 99, top of p. 4](https://ncatlab.org/nlab/show/orientifold#BuchelShiuTye99)). But it is precisely only the cases $N = 2$ and $N = 4$ in which $\mathbb{Z}_N$ may be thought of as being in the D-series, as above.

   By the way, for properly completing the pattern in low degrees, I feel that one should declare the following degenerate cases of binary dihedral / dicyclic groups

   $$2 D_{2} \coloneqq Dic_1 = \mathbb{Z}_4$$

   and

   $$2 D_{0} \coloneqq Dic_0 \coloneqq \mathbb{Z}_2$$

   (I have added these lines to #1 now.)

   Being careful about these degenerate cases seems to be necessary in order to disentangle some subtleties in the string literature. For instance there is originally the idea that one can have a toroidal orientifold of the form

   $$\mathbb{T}^4/\mathbb{Z}_N$$

   for any even $N$, with the reflection being the element $[N/2] \in \mathbb{Z}_N$ (Gimon-Johnson 96, p. 6-7 (7-8 of 32)). But then people find that actually in the case $N = 6$ this leads to inconsistency and they restrict attention to just $N = 2$ and $N = 4$ (Buchel-Shiu-Tye 99, top of p. 4).

   But it is precisely only the cases $N = 2$ and $N = 4$ in which $\mathbb{Z}_N$ may be thought of as being in the D-series, as above.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2019
   • (edited Aug 30th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMore in detail, as we continue the D-series of finite subgroups of $SU(2)$ into the degenerate low range, the cyclic groups $\mathbb{Z}_2$ and $\mathbb{Z}_4$ correspond, as above, to $D1$ and $D3$, but also the case $D2$ appears, as a kind of outlier (not corresponding to a subgroup of $SU(2)$ but of $SU(2) \times SU(2)$): | D-series Dynkin label | finite subgroup of $SU(2)$ | |----------------|------------------------| | $D1 = A1$ | $\mathbb{Z}_2$ | | $D2 = A1 \times A1$ | ($\mathbb{Z}_2 \times \mathbb{Z}_2$) | | $D3 = A3$ | $Dic_1 \simeq \mathbb{Z}_4$ | | $D4$ | $2 D_4 \simeq Dic_2 \simeq Q_8$ | | $D5$ | $2 D_6 \simeq Dic_3 $ | | $D6$ | $2 D_8 \simeq Dic_4 $ | | $\vdots$ | $\vdots$ |

   More in detail, as we continue the D-series of finite subgroups of $SU(2)$ into the degenerate low range, the cyclic groups $\mathbb{Z}_2$ and $\mathbb{Z}_4$ correspond, as above, to $D1$ and $D3$, but also the case $D2$ appears, as a kind of outlier (not corresponding to a subgroup of $SU(2)$ but of $SU(2) \times SU(2)$):

   D-series Dynkin label	finite subgroup of $SU(2)$
   $D1 = A1$	$\mathbb{Z}_2$
   $D2 = A1 \times A1$	($\mathbb{Z}_2 \times \mathbb{Z}_2$)
   $D3 = A3$	$Dic_1 \simeq \mathbb{Z}_4$
   $D4$	$2 D_4 \simeq Dic_2 \simeq Q_8$
   $D5$	$2 D_6 \simeq Dic_3$
   $D6$	$2 D_8 \simeq Dic_4$
   $\vdots$	$\vdots$