Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in terms of the roots $\alpha_{i}$?
9
-
3$\begingroup$ You can use Borel-de Siebenthal theory to find the maximal subroot systems, then induct. I guess at some point you also have to consider maximal parabolic subroot systems as well. $\endgroup$– Sam HopkinsCommented Jun 16, 2022 at 16:03
-
1$\begingroup$ Have you looked in Carter's article on conjugacy classes in Weyl groups? I doubt he explicitly describes exactly what you want, but his computations might be relevant. $\endgroup$– LSpiceCommented Jun 16, 2022 at 16:39
-
1$\begingroup$ @LSpice I have not, I will take a look. Thank you. $\endgroup$– PSHINH2Commented Jun 16, 2022 at 16:44
-
1$\begingroup$ @PSHINH2 note that just extending by the lowest root won't get you all possibilities except for ADE diagrams for the others (BCFG) you need to also allow extending by the lowest short root (see the answer here ) $\endgroup$– CallumCommented Jun 17, 2022 at 12:31
-
1$\begingroup$ For a full classification check out this paper by Oshima which goes through it pretty extensively. $\endgroup$– CallumCommented Jun 17, 2022 at 12:32
|
Show 4 more comments