Schubert calculus and torsion explosion

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.

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

References in corpus (3)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• The two-color Soergel calculus
• A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods

Cited by in corpus (14)

• Soergel calculus and Schubert calculus
• An application of TQFT to modular representation theory
• Modular perverse sheaves on flag varieties I: tilting and parity sheaves
• A quantum version of the algebra of distributions of ${\mathrm{SL}}_2$
• Singularities of Schubert Varieties within a Right Cell
• Soergel bimodules for universal Coxeter groups
• Filtered moment graph sheaves
• On the character of certain modular irreducible representations
• Subquotient categories of the affine category O at the critical level
• Loewy structure of $G_{1}T$-Verma modules of singular highest weights
• Canonical bases for Fock spaces and tensor products
• Jucys-Murphy elements for Soergel bimodules
• The Hard Lefschetz Theorem in positive characteristic for the Flag Varieties
• Parallelotope tilings and $q$-decomposition numbers