Talk:Generalized flag variety

Stiefel manifold

Actually Stiefel manifold does it with orthogonal groups, not general linear groups.

Charles Matthews 11:21, 6 Feb 2005 (UTC)

Parabolic subgroups and contractible

A recent deletion deleted material that was vague, but material that definitely contributed to the article.

It removed the discussion of parabolic subgroups of GL, though flag manifolds are both well understood through their stabilizers and their stabilizers well understood through them. This should be restored (preferably made less vague).

It also removed the statement that the standard flag manifolds are projective varieties. Since this is both a critical property they have and a place where they are critical examples, this should be restored (perhaps less vaguely).

It also removed an interesting principle that the unipotent subgroup is contractible, so in some cases may be ignored. I'm not sure I understood it, but removing a general principle just because it is general seems a very poor idea. This should be restored, and if someone understands it clearly, made less vague. However, it cites specific examples of the principal which are important for the article.

The rest of the recent edits have been good. At one point the phrase "special linear group" appears when probably one means "general linear group" or "automorphism group". No possibility reads that well, and simply deleting the words fixes the problem, but I didn't want to remove meaning just because of a typo. JackSchmidt (talk) 21:09, 26 September 2008 (UTC)[reply]

Hey, I'm glad someone else cares about this article which has been so neglected! But re points 1 and 2, these are both covered properly in the article now, so the deletion eliminated duplication.

For the third point, I am unconvinced that the contractibility of the unipotent subgroup (over R or C) is particularly relevant. In this context the transitive action of a maximal compact subgroup is a much more pertinent observation which packs essentially the same punch. I didn't spot the general/special typo yet. Nilradical (talk) 21:41, 26 September 2008 (UTC)[reply]

wow

this is some amazing stuff but i can't grasp it at all.....bravo you smart bastards! —Preceding unsigned comment added by 173.171.3.145 (talk) 07:31, 29 December 2010 (UTC)[reply]

Thank you! 67.198.37.16 (talk) 06:24, 15 November 2020 (UTC)[reply]

How to represent a "double" or 8 torus?

Hi, i am not really sure where to ask this, but I am looking for a way to represent the 2 dimensional compact manifold that is a double torus, alternatively stated: the connected sum of 2 toruses of dimension 2. I am trying to model something, but would prefer a representation close to grassmanian manifold / flag variety / ... without using an "atlas of charts" cutting the topology up which would force me to model fields in the spaces with seperate functions on each chart...

I remember looking this up a few years ago, and I had found a paper/book in which a variety was constructed which was the double torus, however I cant find it back...

Is there a comprehensive overview somewhere on the web which lists compact manifolds and shows the relations between them ? i.e. I would be able to search by dimension, orientability, choose 2 spaces and take connected sum, or product space? the spaces wouldnt need to display a way of modeling the space, just names and perhaps some properties such that I could identify which space I am envisioning and I can communicate it to others? 11:32, 25 April 2012‎ User:83.134.158.212

I can't quite tell what you are talking about, are you talking about the genus-2 Riemann surface? This thing: Genus g surface? If so, then I imagine most-any book on Riemann surfaces should cover it. It's a fascinating topic, well-worth every minute spent on it. 67.198.37.16 (talk) 06:31, 15 November 2020 (UTC)[reply]