Matrix-Ball Construction of affine Robinson–Schensted correspondence

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.

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References

  1. 1.

    Ariki, S.: Robinson–Schensted correspondence and left cells. arXiv:math/9910117 (1999)

  2. 2.

    Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex classical groups. Math. Ann. 259, 153–199 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29(2), 178–218 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Britz, T., Fomin, S.: Finite posets and Ferrers shapes. Adv. Math. 158(1), 86–127 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Chmutov, M.: Affine Matrix-Ball Construction Java program. https://sites.google.com/site/affiners/ (2015)

  6. 6.

    Fulton, W.: Young tableaux: with applications to representation theory and geometry, Volume 35 of London Mathematical Society Student Texts. Cambridge University Press (1997)

  7. 7.

    Garsia, A.M., McLarnan, T.J.: Relations between Young’s natural and the Kazhdan–Lusztig representations of \(S_n\). Adv. Math. 69(1), 32–92 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Greene, C., Kleitman, D.J.: The structure of Sperner \(k\)-families. J. Comb. Theory Ser. A 20(1), 41–68 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Honeywill, T.: Combinatorics and Algorithms Associated with the Theory of Kazhdan–Lusztig Cells. PhD thesis, University of Warwick (2005)

  10. 10.

    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Lusztig, G.: Cells in affine Weyl groups, IV. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(2), 297–328 (1989)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Pak, I.: Periodic permutations and the Robinson–Schensted correspondence. http://www.math.ucla.edu/pak/papers/inf2.pdf (2003)

  13. 13.

    Shi, J. Y.: Kazhdan–Lusztig cells of certain affine Weyl groups. Volume 1179 of Lecture Notes in Mathematics. Springer (1986)

  14. 14.

    Shi, J.Y.: The generalized Robinson–Schensted algorithm on the affine Weyl group of type \(A_{n-1}\). J. Algebra 139(2), 364–394 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Stanley, R. P.: Enumerative combinatorics, vol. 2. Volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

  16. 16.

    Viennot, G.: Une forme géométrique de la correspondance de Robinson–Schensted. Combinatoire et représentation du groupe symétrique 29–58 (1977)

  17. 17.

    Xi, N.: The Based Ring of Two-Sided Cells of Affine Weyl Groups of Type \(A_{n-1}\). Volume 749 of Memoirs of the American Mathematical Society. American Mathematical Society (2002)

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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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Correspondence to Michael Chmutov.

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In memory of V. A. Yasinskiy.

MC was partially supported by NSF Grants DMS-1148634 and DMS-1503119; PP was partially supported by NSF Grants DMS-1148634, DMS-1351590, and Sloan Fellowship.

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Chmutov, M., Pylyavskyy, P. & Yudovina, E. Matrix-Ball Construction of affine Robinson–Schensted correspondence. Sel. Math. New Ser. 24, 667–750 (2018). https://doi.org/10.1007/s00029-018-0402-6

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Keywords

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

Mathematics Subject Classification

  • 05E10
  • 20C08