TITLE: Geometric proof of Robinson-Schensted-Knuth correspondence?
QUESTION [8 upvotes]: Famous Robinson Schensted Knuth correspondence gives a correspondence between the matrices with non-negative integer entries and pair of semi standard tableaux. The proof that I have seen is highly combinatorial e.g. in Knuth's paper [Permutations, matrices, and generalized young tableaux]. Does there exist a geometrical proof of this correspondence?

REPLY [2 votes]: Not involved approach to RSK is in 
Danilov, V. I.; Koshevoy, G. A. The octahedron recurrence and RSK-correspondence. Sém. Lothar. Combin. 54A (2005/07), Art. B54An, 16 pp. (electronic)