subgroups with the same number of roots that the group. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:33:43Z http://mathoverflow.net/feeds/question/76835 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76835/subgroups-with-the-same-number-of-roots-that-the-group subgroups with the same number of roots that the group. arivero 2011-09-30T08:36:39Z 2011-10-06T17:14:37Z <p>When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10) and its subgroup SU(4)xSU(2)xSU(2), no such removal happens, the Dynkin diagrams have the same number of nodes.</p> <p>How usual is this? For which cases we can find subgroups with the same number of roots (the same number of nodes in the Dynkin diagram, I mean) that the main group? </p> <p>For the aforementionated case, if one tries to remove a further root then the inclusion relationship does not hold anymore: from SO(10) we got to SO(8), but from 4-2-2 we go either to SU(3)xSU(2)xSU(2) or to SU(4)xSU(2), and neither of them are subgroups of SO(8).</p> http://mathoverflow.net/questions/76835/subgroups-with-the-same-number-of-roots-that-the-group/76848#76848 Answer by Shripad for subgroups with the same number of roots that the group. Shripad 2011-09-30T12:25:52Z 2011-09-30T12:25:52Z <p>Removing a random edge and keeping the nodes do not give you a subgroup. The subgroups you mention are obtained by adding one node (and a few edges) and then by removing an inner node. Perhaps you should see Borel-Siebenthal's paper on `maximal subgroups of maximal rank in compact Lie groups' for more on this process. </p> http://mathoverflow.net/questions/76835/subgroups-with-the-same-number-of-roots-that-the-group/77375#77375 Answer by Chuck Hague for subgroups with the same number of roots that the group. Chuck Hague 2011-10-06T17:14:37Z 2011-10-06T17:14:37Z <p>A good modern introduction to the subject is given in Martin Liebeck's survey article "Introduction to the subgroup structure of algebraic groups." It's a chapter in the book "Representations of Reductive Groups." If you have institutional access, you can get the chapter from the Cambridge ebooks website here: <a href="http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511600623&amp;cid=CBO9780511600623A013" rel="nofollow">http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511600623&amp;cid=CBO9780511600623A013</a>.</p> <p>It is also on Google Books here: <a href="http://books.google.com/books?id=t_siS0VHIgAC&amp;pg=PA129&amp;lpg=PA129" rel="nofollow">http://books.google.com/books?id=t_siS0VHIgAC&amp;pg=PA129&amp;lpg=PA129</a></p>