Matrix-Ball Construction of affine Robinson-Schensted correspondence

Abstract

In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, ρ)$ where $P$ and $Q$ are tabloids and $ρ$