Dual Affine Robinson-Schensted Correspondence

TL;DR

This paper introduces a dual affine Robinson-Schensted correspondence that extends classical combinatorial bijections to affine symmetric groups, connecting algebraic, geometric, and combinatorial structures.

Contribution

It generalizes classical growth diagram constructions to the affine case and relates to Kazhdan-Lusztig cells and affine flag varieties.

Findings

Provides a bijection between affine symmetric group elements and tuples involving tabloids and partitions.

Generalizes Fomin's growth diagrams and Viennot's shadow lines to the affine setting.

Parametrizes Kazhdan-Lusztig cells in affine type A.

Abstract

We introduce the dual affine Robinson-Schensted correspondence that gives a bijection between the extended affine symmetric group and tuples $(\bar{P},\bar{Q},\lambda,N)$, where $\bar{P}$ and $\bar{Q}$ are tabloids, $\lambda$ is a partition, and $N$ is an integer, subject to compatibility conditions. The construction generalizes Fomin's growth diagrams and Viennot's shadow lines for the classical Robinson-Schensted correspondence on the symmetric group, and is dual to the affine matrix ball construction as well as Shi's correspondence, in the sense that the $P$-tabloids are the same, and the $Q$-tabloids are related by affine evacuation. As a consequence, our construction also parametrizes Kazhdan-Lusztig cells in affine type $A$. 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.