Affine RSK correspondence and crystals of level zero extremal weight modules

TL;DR

This paper introduces an affine version of the RSK correspondence linking generalized affine permutations to pairs of crystals, extending classical combinatorial bijections to the affine setting with applications to crystal theory.

Contribution

It develops an affine RSK correspondence that maps affine permutations to crystal pairs, preserving crystal structures and generalizing previous affine combinatorial correspondences.

Findings

Affine RSK maps permutations to crystal pairs of the same shape.

The map preserves the crystal equivalence structure.

A dual affine RSK correspondence is also established.

Abstract

We give an affine analogue of the Robison-Schensted-Knuth (RSK) correspondence, which generalizes the affine Robinson-Schensted correspondence by Chmutov-Pylyavskyy-Yudovina. The affine RSK map sends a generalized affine permutation of period $(m,n)$ to a pair of tableaux $(P,Q)$ of the same shape, where $P$ belongs to a tensor product of level one perfect Kirillov-Reshetikhin crystals of type $A_{m-1}^{(1)}$, and $Q$ belongs to a crystal of extremal weight module of type $A_{n-1}^{(1)}$ when $m,n\ge 2$. We consider two affine crystal structures of types $A_{m-1}^{(1)}$ and $A_{n-1}^{(1)}$ on the set of generalized affine permutations, and show that the affine RSK map preserves the crystal equivalence. We also give a dual affine Robison-Schensted-Knuth correspondence.