Schubert calculus and torsion explosion

TL;DR

This paper reveals that certain numerical patterns in Schubert calculus are linked to torsion phenomena in algebraic structures, providing counterexamples to longstanding conjectures in representation theory.

Contribution

It uncovers a connection between Schubert calculus and torsion explosion, leading to counterexamples to Lusztig's and James's conjectures in higher rank cases.

Findings

Numbers in Schubert calculus grow exponentially in higher rank.

Counterexamples to Lusztig's conjecture are constructed.

Counterexamples to James's conjecture are also provided.

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.