Search Results

One classical source for the case $n=2$, with somewhat old-fashioned notation for some of the related groups, is a paper by Richard Brauer and his student Cecil Nesbitt: On the modular characters of g …

Since the comments are already getting long, I'll add this in community-wiki format to clarify a few points. I should emphasize that I'm not at all a specialist in GIT but have dealt with neighborh …

Let me add a couple of things to what grghxy has said. 1) The study of these groups over $\mathbb{Z}$ has been complicated, going back to Chevalley's work and Borel's lecture notes (which aren't quit …

Angelo has indicated the most classical type of counterexample involving two semisimple elements of $\mathrm{PGL}_n$, but the question warrants some further discussion. Though it is posed (and maybe …

The answer to the question might be yes (over an algebraically closed field of any characteristic), though it's not clearly documented in the literature. First let me add a reference to the theorem …

To supplement the answer given by Francois, I'd emphasize that the question is essentially algebraic (over an algebraically closed field of any characteristic) in the spirit of the Borel-Chevalley str …

Thanks for adding some further context. Maybe I can partly answer your question by making a series of comments: 1) At the beginning it's important to specify that the Borel subgroup $B_0$ is $F$-st …

To provide a more balanced context for the question (and the answer by pm), it's useful to separate the elementary notion of BN-pair from the far more sophisticated structure theory of algebraic group …

While waiting for nosr to make the comments here into a full answer, I'll add some references to the literature. The early sources are quite technical and don't provide much in the way of user-frien …

The question itself seems too elementary for this site, since it just involves the standard axiomatic treatment of root systems as in Bourbaki Groupes et algebres de Lie, VI.1.7. The question is rea …

Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give …

There are several questions being asked (and an unexplained reference to a field of definition), but the answer to at least one of them is no: Take $G_1 = G_2 = \mathrm{SL}_3(\mathbb{C})$, with a give …

I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. …

Paul Garrett's book and lecture notes provide a reasonable approach to the subject, for which there are few textbook options. There are of course other lecture notes, usually slanted in some way whi …

EDIT: I didn't comment at first on your actual question, since I wasn't familiar enough with that passage in Borel-Tits. Their (3) strikes me as wrongly stated. Moreover, it doesn't seem to come up …