Timeline for The simple reflections of the Weyl group in $\operatorname{SO}_{2n}(\mathbb C)$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 2, 2019 at 2:11 | comment | added | Ami | @SamHopkins I added it to my question. | |
| Sep 30, 2019 at 16:10 | comment | added | YCor | @LSpice Yes. Let's say $T_{2m}$ the whole upper triangular group, and write $\mathrm{SO}(q)$: then $T_{2m}\cap\mathrm{SO}(q)$ is not Borel in $\mathrm{SO}(q)$ in general, notably for the quite usual forms $q=x_1^2+\dots+x_{2m}^2$ (which usually defines $\mathrm{SO}_{2m}$) or $q=x_1^2+\dots+x_m^2-x_{m+1}^2-\dots-x_{2m}^2$ and $q=x_1x_{m+1}+\dots+x_mx_{2m}$ (which are both standard forms for the split form); however it is for $q=x_1x_{2m}+\dots+x_mx_{m+1}$. | |
| Sep 30, 2019 at 15:55 | comment | added | LSpice | @YCor, just to be clear, you mean that the group of upper-triangular matrices in $\mathrm{SO}_{2n}$ is not Borel unless one takes a particular quadratic form (in which case it is Borel), right? (I suppose one could say equivalently that the group of upper-triangular matrices isn't Borel unless one chooses a basis adapted appropriately to the choice of quadratic form.) | |
| Sep 30, 2019 at 15:23 | answer | added | jorge vargas | timeline score: -3 | |
| Sep 30, 2019 at 14:56 | history | edited | Ami | CC BY-SA 4.0 |
added 24 characters in body
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| Sep 29, 2019 at 20:30 | comment | added | Ami | Yes, you are right, I meant for the torus $T$ to be of the form $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ and $B$ a Borel subgroup containing $T$. | |
| Sep 29, 2019 at 19:40 | review | Close votes | |||
| Oct 11, 2019 at 8:28 | |||||
| Sep 29, 2019 at 19:34 | comment | added | Sam Hopkins | Are you asking, for each $s_i \in W$, what is the $T$-coset in $\mathrm{SO}_{2n}(\mathbb{C})$ corresponding to $s_i$ in the identification $W \simeq N(T)/T$? | |
| Sep 29, 2019 at 17:48 | comment | added | YCor | The group of upper triangular matrices in $SO_{2n}(\mathbf{C})$ is not Borel. One needs to take the quadratic form in a special form, for instance $\sum_{i=1}^nx_ix_{2n+1-i}$. If you're asking for an explicit matrix form, you need to be more specific. Also the maximal torus in a Borel is not unique, so I guess you implicity want it to be defined as the subgroup of diagonal matrices therein. | |
| Sep 29, 2019 at 16:18 | history | asked | Ami | CC BY-SA 4.0 |