I asked Dale Peterson this in '95 and he suggested building the Bott-Samelson instead, then mapping it onto the flag manifold. The original papers of B & S from ~1951 are very nice.

There's a lot to say and I don't know all of it, so I would point you to the book arxiv.org/abs/1301.3569

Here's a '93 paper by Witten attributing the observation to Gepner. '91. arxiv.org/abs/hep-th/9312104

This is indeed the approach that finally got me on board with spectral sequences. It is easy to show that $H^*(Gr_F A^\bullet)$ is typically bigger than the correct answer, and then when you try to correct it... you could have invented spectral sequences. I'd also like to take this opportunity to point out that Dickens' "A Christmas Carol" features an important spectral sequence.

Yeah, nonreducedness is important for me. I'm checking out the Guba-Sapir article math.vanderbilt.edu/sapirmv/ftp/pub/diagramgroups/dg.pdf

Consider a slice (2-dimensional) of the subprincipal nilpotent inside the principal, and take its preimage under the Springer resolution; the fiber over the simple surface singularity will be an ADE Dynkin curve. But what if you start with a non-simply-laced group? Then the subprincipal nilpotent isn't simply connected, and so, you get an action of its pi_1 on the ADE Dynkin diagram. So a non-simply-laced algebra "tells you" how to obtain it from folding. Interestingly, not every possible folding arises this way -- the groups have favored ways they wish to be obtained.

Check out Appendix A of our arxiv.org/abs/1008.3939

I stole the file and stuck it under my web page where it should stay a good long time (link changed in my answer as well).

They're easily derived from the rank 2 cases. When you reflect at a long root, you replace its value by the sum of its neighbors', minus its value. When you reflect at a short root, then when summing over the neighbors you have to weight the long neighbors by 2 or 3. For example in the F_4 example a-b=>c-d, if you reflect the b it becomes a+c-b, but if you reflect the c it becomes 2b+d-c.

If A is integral you could of course blame the first result on Duistermaat-Heckman.

Cauchy-Binet is exactly the statement that $Alt^k(AB) = Alt^k(A)\cdot Alt^k(B)$ where $A$ is $k\times n$ and $B$ is $n\times k$. The matrix $Alt^k(A)$ is a $1\times{n\choose k}$ matrix (or row vector) whereas $Alt^k(B)$ is a column vector, and their product is essentially a dot product.