W-symmetry, topological vertex and affine Yangian

We discuss the representation theory of the non-linear chiral algebra W1+∞ of Gaberdiel and Gopakumar and its connection to the Yangian of û(1) whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of W1+∞ are given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in W1+∞ highest weight representations. Many properties that are difficult to study in the W1+∞ picture turn out to have a simple combinatorial interpretation, once translated to the Yangian picture.