Name for a particular subgroup of parabolic subgroups of the general linear groups. - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T12:32:12Z http://mathoverflow.net/feeds/question/114545 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114545/name-for-a-particular-subgroup-of-parabolic-subgroups-of-the-general-linear-group Name for a particular subgroup of parabolic subgroups of the general linear groups. Hung Nguyen 2012-11-26T16:06:19Z 2012-11-26T21:05:51Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/114544/name-for-a-particular-subgroup-of-parabolic-subgroups-of-the-general-linear-group" rel="nofollow">Name for a particular subgroup of parabolic subgroups of the general linear groups.</a> </p> </blockquote> <p>Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms stabilizing a flag $V=V_1\supset V_2\supset\cdots\supset V_1$ is called a parabolic subgroup of $GL(V)$. I am interested in the subgroup $Q$ of $P$ consisting of all automorphisms $h$ such that the induced automorphism $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the identity for every $i$. My question is: </p> <p><strong>Is this subgroup $Q$ named somewhere yet? If not, can you recommend a name?</strong></p> <p>Similarly, I am also interested in the subgroup $T$ of $P$ consisting of automorphisms $h$ such that the group of induced automorphisms $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the symmetric group (on a basis of $V_i/V_{i+1}$). Is this subgroup $T$ named somewhere yet? If not, can you recommend a name? Is there anyway to realize that $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is a symmetric group without looking a specific basis of $V_i/V_{i+1}$)?</p> <p>Finally, I would appreciate very much if you have any reference on the study of these subgroups.</p> http://mathoverflow.net/questions/114545/name-for-a-particular-subgroup-of-parabolic-subgroups-of-the-general-linear-group/114582#114582 Answer by Nick Gill for Name for a particular subgroup of parabolic subgroups of the general linear groups. Nick Gill 2012-11-26T20:53:31Z 2012-11-26T21:05:51Z <p>As David says $Q$ is the <strong>unipotent radical</strong> of $P$. The subgroup $T$ is a preimage of the <strong>Weyl group</strong> $W$ of the group $G_i\cong GL(V_i/ V_{i+1})$. This group $T$ looks a direct product of $Q$ with a big chunk of a <strong>Levi complement</strong> of $P$. The Levi complement is a direct product isomorphic to $G_1\times\cdots \times G_k$; to obtain the group $T$, you replace the $i$-th factor by the normalizer $N$ of a maximal split torus $T_0$ of $G_i$.</p> <p>This is, in fact, the typical way to realize the Weyl group of $G_i$ -- $W$ is isomorphic to the quotient $N/T_0$ -- but this is effectively the same thing as your method of fixing a specific basis of $V_i/V_{i+1}$. The Weyl group rears its head in lots of different ways (most especially as a Coxeter group related to the Dynkin diagram of $G_i$) so this is certainly not the only way to realise it. I don't, however, see any other way to realise your group $T$ (although it depends what you mean by `realise'!).</p> <p>As for references, it depends on what kind of approach you want. If you want a treatment of $GL_n$ as an algebraic group then I recommend anything by Carter or Humphreys, or else there is the book by Borel. All of these people work in much greater generality than $GL_n$ though. If you just want to understand $GL_n$, then standard algebra texts like the one of Jacobson might be your best bet. (I have e-copies of some of these. If you want them, email me.)</p>