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Selecta Mathematica

, Volume 24, Issue 2, pp 667–750 | Cite as

Matrix-Ball Construction of affine Robinson–Schensted correspondence

  • Michael Chmutov
  • Pavlo Pylyavskyy
  • Elena Yudovina
Article
  • 34 Downloads

Abstract

In his study of Kazhdan–Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson–Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi’s algorithm. As a byproduct, we also give a way to realize the affine correspondence via the usual Robinson–Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between the extended affine symmetric group and collection of triples \((P, Q, \rho )\) where P and Q are tabloids and \(\rho \) is a dominant weight. The weights \(\rho \) get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

Keywords

Affine Weyl group Kazhdan–Lusztig cells Matrix-Ball Construction Robinson–Schensted correspondence 

Mathematics Subject Classification

05E10 20C08 

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Notes

Acknowledgements

The authors are grateful to Joel Lewis and an anonymous referee for reading the paper and providing valuable feedback, and to Darij Grinberg for helpful comments.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Michael Chmutov
    • 1
  • Pavlo Pylyavskyy
    • 1
  • Elena Yudovina
    • 1
  1. 1.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

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