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 byproduct, 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 the extended affine symmetric group and collection of triples (Formula presented.) where P and Q are tabloids and (Formula presented.) is a dominant weight. The weights (Formula presented.) get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.
- Affine Weyl group
- Kazhdan–Lusztig cells
- Matrix-Ball Construction
- Robinson–Schensted correspondence