Recent progress in algebraic combinatorics

Source abstract

We survey three recent breakthroughs in algebraic combinatorics. The first is the proof by Knutson and Tao, and later Derksen and Weyman, of the saturation conjecture for Littlewood-Richardson coefficients. The second is the proof of the n ! n! and ( n + 1 ) n − 1 (n+1)^{n-1} conjectures by Haiman. The final breakthrough is the determination by Baik, Deift, and Johansson of the limiting behavior of the length of the longest increasing subsequence of a random permutation.