Affine RSK correspondence and crystals of level zero extremal weight modules

2022/03/01 by Kwon, Jae-Hoon, Lee, Hyunse
#05E10 #17B37 #22E46 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2203.00406

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≥ 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.

Related

• On Combinatorial Models for Affine Crystals
• Demazure crystals of generalized Verma modules and a flagged RSK correspondence
• Categorifying the tensor product of the Kirillov-Reshetikhin crystal B^1,1 and a fundamental crystal
• Simplicity and similarity of Kirillov-Reshetikhin crystals
• Some results on the crystal commutor and affine sl(n) crystals