Timeline for What is Borel-de Siebenthal theory?
Current License: CC BY-SA 2.5
5 events
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| Mar 23, 2011 at 13:12 | comment | added | Jeffrey Adams | Expanding a little on Allen's algorithm, if you remove a single node with prime label, or two nodes with label 1, from the extended Dynkin diagram $\tilde D$, then the resulting subgroup H is maximal. Furthermore (modulo the action of $Z_{sc}$ on the extended Dynkin diagram) this classifies the maximal, reductive subgroups of the same rank as G up to conjugacy. Then you get every reductive, equal rank subgroup by iterating. However I do not know any way to keep track of conjugacy classes of the resulting subgroups. | |
| Jun 24, 2010 at 1:48 | comment | added | Allen Knutson | Steps 1+2 together replaces a root system by a sub root system. (Grr. I don't want to call it a sub-root system or a root subsystem. Anyway...) If you remove more than one vertex at step 2, the subsystem will be of lower rank, corresponding to a Levi subgroup of what you'd get from removing just one. E.g., if you start with G semisimple and remove all the vertices at step 2, the result is the *empty) Dynkin diagram of a maximal torus of G. As for the other remark, if you affinize and remove 1 vertex from an A_m diagram, you've done nothing. You're stuck; no more maximal rank s.s. subgroups. | |
| Jun 23, 2010 at 15:56 | comment | added | Jim Humphreys | P.S. I don't follow the parenthetic remarks in Allen's numbered steps, about staying semisimple. (And change "is" to "are" in my first line. Though I ran out of characters.) | |
| Jun 23, 2010 at 15:51 | comment | added | Jim Humphreys | I should have emphasized that the extended/affine diagrams and vertex removal is basic to the Borel-de Siebenthal classification of maximal subgroups, as well as some related developments: classification of maximal subgroups of semisimple algebraic groups or finite groups of Lie type in prime characteristic (Liebeck-Seitz, expanding on Dynkin's 1950s papers); centralizers of semisimple elements and pseudo-Levi subgroups or subalgebras (Carter-Deriziotis, Sommers); probably some of the geometry of affine Kazhdan-Lusztig cells, related in Lusztig's bijection to unipotent classes of dual groups.. | |
| Jun 23, 2010 at 13:46 | history | answered | Allen Knutson | CC BY-SA 2.5 |