Using row insertion or jeu-de-taquin, one can define a product on the set of semistandard Young tableaux. This product is not commutative, but given two tableaux T_1,T_2 of rectangular shape, there is a unique pair of tableaux T_2',T_1', such that T_i' has the same shape as T_i, and T_1T_2 = T_2'T_1'. The map from T_i to T_i' is called the combinatorial R-matrix.

In this talk, I'll define the product, and explain how it is closely connected to the Littlewood-Richardson rule. I'll use this connection to show the existence of the combinatorial R-matrix, and we'll see why the assumption of rectangular shape is necessary. Then I'll explain how the combinatorial R-matrix can be used to study a cellular automaton called the Box-Ball System. Speaker(s): Gabriel Frieden (University of Michigan)