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Abstract: 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 Robinson–Schensted correspondence into the affine type A setting have been studied. Prominent examples include Shi’s insertion algorithm and the affine matrix ball construction by Chmutov–Pylyavskyy–Yudovina. In particular, using the affine matrix ball construction, Boixeda–Ying–Yue showed that the two-sided cells and S-cells agree when the nilpotent is of rectangular type.

In this talk, 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, and geometrically natural in terms of relative positions of affine flags.

Ongoing joint work with Sylvester Zhang.