Schubert calculus and torsion explosion

Abstract

The author observes that certain numbers occurring in Schubert calculus for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="SL Subscript n"> <mml:semantics> <mml:msub> <mml:mtext>SL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\text {SL}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="SL Subscript n"> <mml:semantics> <mml:msub> <mml:mtext>SL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\text {SL}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over fields of positive characteristic. The examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.