Schubert calculus and torsion explosion
Representation Theory
2016-09-15 v3
Abstract
We observe that certain numbers occurring in Schubert calculus for SL_n also occur as entries in intersection forms controlling decompositions of Soergel bimodules and parity sheaves in higher rank. These numbers grow exponentially. This observation gives many counterexamples to Lusztig's conjecture on the characters of simple rational modules for SL_n over a field of positive characteristic. We explain why our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.
Cite
@article{arxiv.1309.5055,
title = {Schubert calculus and torsion explosion},
author = {Geordie Williamson},
journal= {arXiv preprint arXiv:1309.5055},
year = {2016}
}
Comments
23 pages, v2: proofs contain more detail, added an appendix with Kontorovich and McNamara proving exponential growth of torsion. v3: final version, to appear in JAMS