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Schubert calculus and torsion explosion


Author: Geordie Williamson; with a joint appendix with Alex Kontorovich; Peter J. McNamara
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 20C20, 20G05; Secondary 14N15, 14M15
DOI: https://doi.org/10.1090/jams/868
Published electronically: October 20, 2016
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Abstract: The author observes that certain numbers occurring in Schubert calculus for $ \text {SL}_n$ also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig's conjecture on the characters of simple rational modules for $ \text {SL}_n$ over fields of positive characteristic. The examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.


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Additional Information

Geordie Williamson
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany
Email: geordie@mpim-bonn.mpg.de

Alex Kontorovich
Affiliation: Rutgers University, New Brunswick, New Jersey
Email: alex.kontorovich@rutgers.edu

Peter J. McNamara
Affiliation: University of Queensland, Brisbane, Queensland, Australia
Email: p.mcnamara@uq.edu.au

DOI: https://doi.org/10.1090/jams/868
Received by editor(s): June 22, 2015
Received by editor(s) in revised form: May 5, 2016, and August 4, 2016
Published electronically: October 20, 2016
Additional Notes: The first author of the Appendix is partially supported by an NSF CAREER grant DMS-1455705 and an Alfred P. Sloan Research Fellowship.
Dedicated: Dedicated to Meg and Gong.
Article copyright: © Copyright 2016 American Mathematical Society