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Affine RSK Correspondence and Crystals of Level Zero Extremal Weight Modules

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Abstract

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 \(A_{m-1}^{(1)}\), and Q belongs to a crystal of extremal weight module of type \(A_{n-1}^{(1)}\) when \(m,n\geqslant 2\). 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.

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Acknowledgements

The authors would like to thank Dongkwan Kim for the kind explanation on the works [2, 4] and the referees for very careful reading and helpful comments.

Funding

This work is supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2019R1A2C1084833 and 2020R1A5A1016126).

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Authors and Affiliations

  1. Department of Mathematical Sciences and RIM, Seoul National University, Seoul, 08826, Korea

    Jae-Hoon Kwon

  2. Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Hyunse Lee

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  1. Jae-Hoon Kwon
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Correspondence to Hyunse Lee.

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Kwon, JH., Lee, H. Affine RSK Correspondence and Crystals of Level Zero Extremal Weight Modules. Transformation Groups (2024). https://doi.org/10.1007/s00031-024-09857-0

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