Schubert calculus and torsion explosion

Williamson, Geordie

doi:10.1090/jams/868

Schubert calculus and torsion explosion
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by Geordie Williamson; with a joint appendix with Alex Kontorovich; with a joint appendix with Peter J. McNamara

J. Amer. Math. Soc. 30 (2017), 1023-1046

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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