Matrix-Ball Construction of affine Robinson-Schensted correspondence

TL;DR

This paper extends the Matrix-Ball Construction to affine Robinson-Schensted correspondence, providing a combinatorial realization, a connection to the usual bumping algorithm, and a symmetry property of the inverse fibers.

Contribution

It generalizes the Matrix-Ball Construction to affine type, linking it with the classical bumping algorithm and establishing Weyl group symmetry in the affine setting.

Findings

Extended affine Robinson-Schensted correspondence via combinatorial construction

Realized affine correspondence through classical bumping algorithm

Proved Weyl group symmetry in inverse fibers

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 $(P, Q, \rho)$ where $P$ and $Q$ are tabloids and $\rho$ is a dominant weight. The weights $\rho$ 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.