Timeline for Pairing half the sum of the roots with a simple coroot
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2018 at 19:51 | comment | added | Sam Hopkins | The fundamental weights don’t add in that manner because the fundamental weights of a sub root system belong to the span of that sub root system. In general the fundamental weights of a sub root system are not even integral weights of the bigger root system. | |
| Aug 1, 2018 at 19:47 | comment | added | D_S | Say $\Delta = \theta_1 \cup \theta_2$, let $[\theta_i]$ be the root system with base $\theta_i$, and $\rho_i$ half the sum of positive roots in $[\theta_i]$. What I'm confused about is that we certainly don't have $\rho_{\Delta} = \rho_{\theta_1} + \rho_{\theta_2}$, yet the fundamental weights should add in this manner. | |
| Aug 1, 2018 at 19:40 | comment | added | Sam Hopkins | The half sum of positive roots with alpha coordinate zero is the Weyl vector of the maximal parabolic I am taking about. | |
| Aug 1, 2018 at 19:39 | comment | added | Sam Hopkins | No, it’s as I said: your rho is the Weyl vector of the whole root system minus the Weyl vector of the maximal parabolic corresponding to alpha (I.e., the root system generated by all simple roots other than alpha). | |
| Aug 1, 2018 at 19:34 | comment | added | D_S | Wouldn't that make $\rho$ equal to Weyl vector of the root system corresponding to the singleton $\alpha$? That doesn't seem right. | |
| Aug 1, 2018 at 19:26 | comment | added | Sam Hopkins | This is true because the Weyl vector is the half sum of positive roots and also equal to the sum of fundamental weights. The rho you defined is just the Weyl vector of the whole root system Phi minus the Weyl vector of the maximal parabolic root system corresponding to alpha. | |
| Aug 1, 2018 at 19:11 | history | asked | D_S | CC BY-SA 4.0 |