What are Flag Manifolds?

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I came across the term while reading, and all the websites I udually frequent for math things lesve me more confused.

In: Mathematics

This won’t be “LI5” because it is a technical object. A Flag Manifold is a geometric object where every point is a “Flag”. A “Flag” is an increasing sequence of vector spaces. See [Wikipedia for more](https://en.wikipedia.org/wiki/Generalized_flag_variety).

A circle is a Flag Manifold. To see this, note that every point on the circle corresponds to the slope of a unique line. The point on the circle with angle T corresponds to the line with slope m=tan(T/2). If we let (A,B) be a point on the plane with slope m, then the vector space generated by this point will be a line with slope m, call it V(m). Therefore, every point on the circle corresponds to the sequence of vector spaces

* 0 < V(m) < R^(2)

And so the circle is a Flag Manifold. This is a simple example, and they get crazy really quickly, because they can encode a lot of information. If you have a N-dimensional space, then there are tons of ways you can make a sequence of vector spaces

* V*_1_* < V*_2_* < V*_3_* <V*_4_*,…, V*_k_*

But, overall, they’re pretty nice compared to other kinds of manifolds and they’re useful in both pure math and physics, as each point contains a lot of information.