The Robinson--Schensted (RS) correspondence admits diagrammatic interpretations via Fomin's growth diagrams and Viennot's shadow line construction. Works of Spaltenstein, Springer, Steinberg, and van Leeuwen connected this combinatorial construction to the relative position map of Springer fibers. Motivated by Kazhdan--Lusztig cell theory, various generalizations of the RS correspondence into the affine type A setting have been studied. Prominent examples include Shi's insertion algorithm and the affine matrix ball construction.
We introduce a new combinatorial construction of the affine RS correspondence via growth diagrams and shadow lines that is in a sense dual to Shi's insertion and the affine matrix ball construction. We conjecture that the growth diagrams we construct admit a natural geometric realization in terms of relative positions of affine flags, similar to the interpretation given by Steinberg and van Leeuwen in the classical case.
Joint with Sylvester Zhang.