Slides and Notes

#1.

Various rings-with-bases, including H**(Fl(1..k; oo)) and Rep_alg(GL(k))

Literary conventions on Schubert varieties:
Bruhat order, B orbits vs. B_- orbits, X_w & X^v
Permutations vs. strings. For Gr(k,n) we use strings with content 0^k 1^{n-k}

String: as we read through the opposite base flag, what's the smallest
of our subspaces that increases in intersection? Draw P^2 and Fl(3) examples.

Recall duality theorem: use that to get a symmetric statement, + Gr duality

ORACLE '96: there is a way of calculating LR coefficients using
tilings, with the strings placed around the boundary.
So what are they? Discover them.

Basic rules for filling. S_0101^2. Combinatorial facts.

#2.

S_0101^4. K-theory. 2-step puzzles.

Equivariant intersection theory: [0]^2 in CC^n.
Restriction to fixed points. Kirwan injectivity for P(CC_a + CC_b).
Fundamental relation: [0]-[oo] = b-a.

No Z_3 symmetry. y_i-y_j factors.
Kirwan injectivity.

#3.

Localization.
The Bott-Samelson crank.

r_alpha int [X_w] [v]
= int (r_alpha.[X_w]) [r_alpha v]
= int ([X_w] - alpha partial_alpha [X_w]) [r_alpha v]
= int ([X_w] - alpha [X_{r_alpha w}]) [r_alpha v]

[X_w]|_v
= r_alpha [X_w]|_{r_alpha v} + alpha r_alpha [X_{r_alpha w}]|_{r_alpha v}

AJS/Billey.
Fulton's isomorphism, and AJS/Billey => pipe dream formula.

R-matrices.
Approaching a proof of the puzzle formula.

Descents.
Separated descent.
Euler characteristics.