User p-samuel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:22:09Z http://mathoverflow.net/feeds/user/26032 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108792/irreducibility-of-fundamental-weyl-modules Irreducibility of fundamental Weyl modules P-Samuel 2012-10-04T09:00:22Z 2012-10-06T20:53:52Z <p>It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the fundamental weights are irreducible if the group is either $SL_n$ or $SO_n$ (Wong 71). For the symplectic group, this is not true for an arbitrary fundamental weight. Foulle provides certain conditions on the weights satisfying which the corresponding Weyl modules of the symplectic group are irreducible. Does one know anything about the irreducibility of the fundamental Weyl modules for the exceptional groups? I believe its true for $G_2$ by results of Premet, Humphreys and others. Is it true also for other exceptional groups?</p> http://mathoverflow.net/questions/105781/failure-of-jacobson-morozov-in-positive-characteristics Failure of Jacobson Morozov in positive characteristics P-Samuel 2012-08-29T01:16:58Z 2012-08-30T19:30:41Z <p>The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the characteristic of the underlying field (assumed to be algebraically closed). This restriction is also required for the "uniqueness" of the triple, up to $C_G(e)$-action. (This result is due to Kostant). In his 1980 paper, Pommerening had removed the restriction on the characteristic in Jacobson-Morozov's theorem, up to very small exceptions (<em>i.e.</em>,characteristic is "bad"). Does the uniqueness as in Kostant's result also hold with this weaker restriction? If it does, then where does Jacobson-Morozov along with uniqueness result of Kostant fail in positive characteristics?</p> http://mathoverflow.net/questions/108792/irreducibility-of-fundamental-weyl-modules/108811#108811 Comment by P-Samuel P-Samuel 2012-10-05T07:41:11Z 2012-10-05T07:41:11Z Thank you. This was indeed very helpful. http://mathoverflow.net/questions/105781/failure-of-jacobson-morozov-in-positive-characteristics/105935#105935 Comment by P-Samuel P-Samuel 2012-08-31T04:54:59Z 2012-08-31T04:54:59Z The example partially answers my next query but I am still curious to know if there is any idea that one can have about the number of orbits of $sl_2$-triples that are mapped to a single nilpotency class, atleast in certain special characteristics, of course apart from the characteristics wehere Kostant's proof works?