Journal of Algebraic Combinatorics

, Volume 50, Issue 1, pp 49–72 | Cite as

Combinatorial wall-crossing and the Mullineux involution

  • Panagiotis Dimakis
  • Guangyi YueEmail author


In this paper, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition (n). As corollaries we explicitly describe the quotients of the partitions which arise in this process. We also prove that the one-row partition is the unique partition that stays regular at any step of the wall-crossing transformation.


Combinatorial wall-crossing Column regularization Monotonicity 



The authors would like to thank Roman Bezrukavnikov for suggesting this project to us and continuous discussions and help throughout the whole process. Also, the authors are grateful to Ivan Losev and to Galyna Dobrovolska for many discussions and to Seth Shelley-Abrahamson for useful revision suggestions.


  1. 1.
    Anno, R., Bezrukavnikov, R., Mirkovic, I.: Stability conditions for slodowy slices and real variations of stability. Mosc. Math. J. 15(2), 187–203 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beilinson, A., Ginzburg, V.: Wall-crossing functors and \(\cal{D}\)-modules. Represent. Theory Am. Math. Soc. 3(1), 1–31 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bezrukavnikov, R., Losev, I.: On Dimension Growth of Modular Irreducible Representations of Semisimple Lie Algebras (2017). arXiv preprint arXiv:1708.01385
  4. 4.
    Bessenrodt, C., Olsson, J.B.: On residue symbols and the Mullineux conjecture. J. Algebr. Comb. 7(3), 227–251 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bessenrodt, C., Olsson, J.B., Xu, M.: On properties of the Mullineux map with an application to Schur modules. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 126, pp. 443–459. Cambridge University Press, Cambridge (1999)Google Scholar
  6. 6.
    Ford, B., Kleshchev, A.S.: A proof of the Mullineux conjecture. Math. Z. 226(2), 267–308 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Haiman, M.: Combinatorics, symmetric functions, and Hilbert schemes. Curr. Dev. Math. 39–111, 2002 (2002)Google Scholar
  8. 8.
    Halacheva, I., Kamnitzer, J., Rybnikov, L., Weekes, A.: Crystals and Monodromy of Bethe Vectors (2017). arXiv preprint arXiv:1708.05105
  9. 9.
    James, G., Kerber, A.: The representation theory of the symmetric group, Encyclopedia of mathematics and its applications, vol. 16. Addison-Wesley Publishing Co., Reading, Mass (1981). With a foreword by P.M. Cohn, With an introduction by Gilbert de B. RobinsonGoogle Scholar
  10. 10.
    Kleshchev, A.S.: Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineux. J. Lond. Math. Soc. 54(1), 25–38 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Losev, I.: Cacti and Cells (2015). arXiv preprint arXiv:1506.04400
  12. 12.
    Losev, I.: Supports of Simple Modules in Cyclotomic Cherednik Categories O (2015). arXiv preprint arXiv:1509.00526
  13. 13.
    Mathas, A.: Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, vol. 15. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  14. 14.
    Walker, G.: Modular Schur functions. Trans. Am. Math. Soc. 346(2), 569–604 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Walker, G.: Horizontal partitions and Kleshchev’s algorithm. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 120, pp. 55–60. Cambridge University Press, Cambridge (1996)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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