Rank one summands of Frobenius pushforwards of line bundles on G/P

2025/07/28 by Rączka, Feliks
#13A35 #14F06 #14G17 #14M15 #20G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.20759

Abstract

Let X=G/P be a partial flag variety, where G is a semi-simple, simply connected algebraic group defined over an algebraically closed field K of positive characteristic. Let F\colon X→ X be the absolute Frobenius morphism. Given a line bundle \mathscrL on X and an integer r≥1, we describe all line bundles that are direct summands of the pushforward F_*^r\mathscrL. For \mathscrL corresponding to a dominant weight, we also compute, for r sufficiently large, the multiplicity of \mathscrO_X as a summand of F_*^r\mathscrL. As an application we answer a question of Gros-Kaneda about the multiplcity of \mathscrL(-ρ) as a direct summand of F_*\mathscrO_G/B.

Citations

• Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications
• The Frobenius morphism in invariant theory II
• The Frobenius morphism in invariant theory
• A characterization of toric varieties in characteristic p
• Frobenius Push-Forwards on Quadrics

Related

• Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications
• Frobenius pushforwards of of vector bundles on projective spaces
• The Frobenius morphism on flag varieties, II
• The Frobenius morphism on flag varieties, I
• A note on the Frobenius morphism on toric varieties