Timeline for Which reflection groups can be enlarged?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Aug 4, 2021 at 0:08 | comment | added | Daniel Sebald | What about prismatic (reducible) groups? In particular, what about $A_1A_{14}$? | |
| Feb 11, 2021 at 22:38 | history | became hot network question | |||
| Feb 11, 2021 at 16:17 | answer | added | Will Sawin | timeline score: 8 | |
| Feb 11, 2021 at 16:04 | comment | added | Nathan Reading | @M.Winter: You reference the answer about $E_8$ to suggest that, e.g. $B_5$ can't be enlarged. If we believe that argument, isn't it easy to also see that, say, $B_3$ can't be enlarged? In the notation of that answer, a reflection group $G$ in $R^3$ containing both $B_3$ and $H_3$ can't be finite. (There is no containment relation between them, and $H_3$ is the largest in the classification.) | |
| Feb 11, 2021 at 15:59 | comment | added | Theo Johnson-Freyd | @Sam Yes, $\mathrm{Weyl}(F_4) = \mathrm{Weyl}(D_4) \rtimes S_3$ | |
| Feb 11, 2021 at 15:43 | comment | added | M. Winter | @DanielSebald Have I understood you correctly: $A_4^*,D_4\subset H_4$, and $A_8^*,D_8\subset E_8$? And let me also ask: we do not have $A_6^*\subset E_6$ or $D_7\subset E_7$? | |
| Feb 11, 2021 at 15:34 | comment | added | M. Winter | @YCor Yes, I replaced that. | |
| Feb 11, 2021 at 15:34 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Feb 11, 2021 at 15:33 | comment | added | YCor | is "finite point group" jargon for "finite subgroup" in this context? | |
| Feb 11, 2021 at 15:27 | comment | added | Daniel Sebald | After rereading the question, I realized that it would be helpful to note that $D_4$ but not $BC_4$ is contained in $H_4$, and $D_8$ but not $BC_8$ is contained in $E_8$. | |
| Feb 11, 2021 at 15:21 | comment | added | LSpice | If we call the long simple roots of $\mathsf F_4$ $\alpha_1$ and $\alpha_2$, and the short simple roots $\alpha_3$ and $\alpha_4$, with $\alpha_2$ adjacent to $\alpha_3$, then there's an order-3 symmetry that fixes $\alpha_1$ and sends $\alpha_2$ to $\alpha_2 + 2\alpha_3$, $\alpha_3$ to $\alpha_4$, and $\alpha_4$ to $-\alpha_3 - \alpha_4$ (I think!). I don't know if that's in $\operatorname W(\mathsf F_4)$. | |
| Feb 11, 2021 at 15:16 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Feb 11, 2021 at 15:06 | comment | added | Daniel Sebald | 1. $A_4 ^*$, $A_7 ^*$, and $A_8 ^*$ can be enlarged into $H_4$, $E_7$, and $E_8$, respectively. | |
| Feb 11, 2021 at 14:52 | comment | added | M. Winter | @LSpice Uhm, I think of the additional symmetries as permuting the generating mirrors (which is possible, given the additional symmetry of the Coxeter-Dynkin diagram). But I suppose that this extended symmetry group is also the full symmetry group of the $F_4$ root system with all vectors of the same length. | |
| Feb 11, 2021 at 14:52 | comment | added | Sam Hopkins | @LSpice: Yes, the Coxeter diagram (unlike the Dynkin diagram) does not have arrows, so has an order two symmetry. | |
| Feb 11, 2021 at 14:48 | comment | added | LSpice | What are the additional symmetries of $\mathsf F_4$? Do they come from ignoring root lengths? | |
| Feb 11, 2021 at 14:46 | comment | added | Sam Hopkins | I'm not sure... | |
| Feb 11, 2021 at 14:43 | comment | added | M. Winter | @SamHopkins Might this lead to its inclusion in $F_4$? | |
| Feb 11, 2021 at 14:42 | comment | added | Sam Hopkins | $D_4$ has an exceptional triality symmetry, for what it's worth. | |
| Feb 11, 2021 at 14:38 | history | asked | M. Winter | CC BY-SA 4.0 |