Combinatorial wall-crossing and the Mullineux involution
- 34 Downloads
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.
KeywordsCombinatorial 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.
- 3.Bezrukavnikov, R., Losev, I.: On Dimension Growth of Modular Irreducible Representations of Semisimple Lie Algebras (2017). arXiv preprint arXiv:1708.01385
- 7.Haiman, M.: Combinatorics, symmetric functions, and Hilbert schemes. Curr. Dev. Math. 39–111, 2002 (2002)Google Scholar
- 8.Halacheva, I., Kamnitzer, J., Rybnikov, L., Weekes, A.: Crystals and Monodromy of Bethe Vectors (2017). arXiv preprint arXiv:1708.05105
- 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
- 11.Losev, I.: Cacti and Cells (2015). arXiv preprint arXiv:1506.04400
- 12.Losev, I.: Supports of Simple Modules in Cyclotomic Cherednik Categories O (2015). arXiv preprint arXiv:1509.00526