Timeline for Subgroups of $W(E_8)$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 3, 2021 at 23:07 | comment | added | André Henriques | @LSpice Sometimes you need to do the trick of passing the the extended Dynkin diagram more than once: for example $A_2\times A_2\times A_2\times A_2$ sits inside $E_8$. | |
| Apr 2, 2021 at 2:36 | comment | added | LSpice | @AndréHenriques's answer is an example of Borel–de Siebenthal theory: maximal-rank subgroups are classified by sub-root systems of the extended root system $\widetilde{E_8}$. Deleting $\alpha_2$ (in the Bourbaki notation) from that extended system gives $A_8$. | |
| Apr 1, 2021 at 19:57 | comment | added | André Henriques | @YCor. Here's how you embed the $A_8$ root system into the $E_8$ root system. Consider the $8$ simple roots of $E_8$ together with the negative of the highest root. Together, they form a copy of the affine $E_8$ Dynkin diagram. Now, there's an obvious copy of the $A_8$ Dynkin diagram inside the affine $E_8$ Dynkin diagram. The same method produces lots of same-rank inclusions between Lie algbras. For example $A_4\times A_4$ sits inside $E_8$ . | |
| Apr 1, 2021 at 18:58 | comment | added | Theo Johnson-Freyd | Equivalent to @WillSawin's answer: $E_8$ Lie group contains an element of order $3$ whose centralizer is $SU(9)/\mathbb{Z}_3$. | |
| Apr 1, 2021 at 18:46 | answer | added | Will Sawin | timeline score: 11 | |
| Apr 1, 2021 at 18:03 | comment | added | Daniel Sebald | There is a flag-transitive compound of 1920 8-simplices with the same vertex arrangement as the 1_42 polytope. | |
| Apr 1, 2021 at 18:02 | comment | added | Will Sawin | @YCor The lattice of vectors in $ ( (1/3) \mathbb Z)^9$ which sum to $0$ and are congruent mod $1$ is a model of $E_8$, and its symmetry group contains $W(E_8) = S_9$ permuting the entries. | |
| Apr 1, 2021 at 17:56 | comment | added | YCor | Out of curiosity, how is $W(A_8)$ embedded into $W(E_8)$? (there's no obvious graph embedding) | |
| Apr 1, 2021 at 17:33 | history | asked | Daniel Sebald | CC BY-SA 4.0 |