CAG@ICH#47

[crazilla]

exemplifying generalization

As Dr. Green amusingly pointed out in Algebraic Geometry, one of the most common things to do in mathematics is generalize - that is, loosen the restrictions of a definition of a concept without trivializing it. A great string of exemplary generalizations begins with the projective line P1.

Now we may think of P1 (ahem, that's P1) as the set of one-dimensional subspaces of a two-dimensional vector space V. (For all we care V = R2.) We can identify a subspace by a vector that generates it, for instance [v] = [v1,v2], with the understanding that parallel vectors identify the same subspace. But that's another story. Our story is the following string of generalizations:

• Projective n-space Pn, thought of as the set of one-dimensional subspaces of an (n + 1)-dimensional vector space. It's n-dimensional because we're "modding out" the dimension of "length" and keeping only "direction". We get the projective line back by setting n = 1.
• The Grassmannian Grk,n, thought of as the set of k-dimensional subspaces of an n-dimensional vector space. It ends up being ("n-choose-k" - 1)-dimensional* because a k-plane can "tilt" by a certain angle relative to each of the n-choose-k coordinate k-planes (except that we can always determine one angle from the rest). For instance, Gr2,3 consists of planes in three-space, each of which has a descriptive angle with the xy-plane, another with the xz-plane, and another with the yz-plane. (Think of beginning with the xy-plane, then using two rotations - one about the x-axis and the other about the z-axis - to get the plane you want.) We get projective n-space back by setting k = 1.
• A partial flag variety Fl(k1, . . . ,kl), thought of as the set of nested subspaces (V1, . . . ,Vl), each Vi of dimension ki and contained inside the following Vi + 1. Let's not get into dimension, but note that this thing is gigantic: Think of including the indices ki one at a time, and you realize that each step increases the dimension by about that of a Grassmannian. (We're "fitting" a new subspace between two others in order, for instance choosing a plane containing a given line and contained in a given space.) We get a Grassmannian back by setting l = 1, i.e. looking at only one "nested" subspace.
• A Bott-Samelson variety BS(P), thought of as the set of subspaces with certain ones nested in certain others, so that the containment lattice looks like a desired (finite connected) poset (with maximum and minimum) P. A "poset" is a partially ordered set, for instance the set of points P = {(0,0),(0,1),(1,0),(1,1)} where one point is considered "bigger" than another if it is farther away from (0,0). (Note that (0,1) and (1,0) are incomparable.) So the (smallest) Bott-Samelson variety BS(P) is the set of quadruples (0,L1,L2,V), where V is a two-dimensional space, 0 is the zero vector, and L1 and L2 are one-dimensional subspaces (lines through 0). (They might be the same line, or they might not.) Finite posets can get pretty interesting on their own, and the dimension of BS(P) is as big as the total dimensions of a new Grassmannian for each point in P. We get a partial flag variety back by asking P to be totally ordered.

What might come next? Well, if we disconnect the poset we just get unions of separate Bott-Samelson varieties, and it's hard to imagine branching out in new dimensions without sounding silly. We could look at the set of strings of Grassmannians under containment, but ultimately this just gives us a subvariety of a ridiculously large Bott-Samelson. But then, Grassmannians themselves are subvarieties of moderately large projective spaces, so perhaps this is worth looking at. More subtly, since we now have definitions of "fractional-dimensional objects", could we extend the idea of a partial flag variety to include some non-integer (but rational) ki, just as we extend the integers to get the rationals? This would allow us to sensibly talk about Bott-Samelson varieties over infinite posets (in a very loose sense of "sensible", of course). We'll see if any such things come (or have come) to pass.

exemplifying everything

One thing our proposed Maths Dept. Wiki would make feasible is an all-access repository of good examples . . . and examples of anything - cumulative exercises for basic Calculus, curiosities that we come up with on napkins, all manner of tableau insertion rules, and so on. Shouldn't there be a massive database of such things online somewhere already? I'd like to be able to do a search for "schubert AND intersection" and get back a bunch of Schubert calculus problems to work or follow through to build my understanding of the basics, then move on in whatever direction i want. Math Examples Wiki. You saw it here first, folks.

* Please petition LiveJournal to adopt html math.