Affine RSK Correspondence and Crystals of Level Zero Extremal Weight Modules

We give an affine analogue of the Robinson-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 \(A_{m-1}^{(1)}\) $A_{m-1}^{(1)}$ , and Q belongs to a crystal of extremal weight module of type \(A_{n-1}^{(1)}\) $A_{n-1}^{(1)}$ when \(m,n\geqslant 2\) $m,n⩾2$ . We consider two affine crystal structures of types \(A_{m-1}^{(1)}\) $A_{m-1}^{(1)}$ and \(A_{n-1}^{(1)}\) $A_{n-1}^{(1)}$ 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.