Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums

2017/12/31 by Valentina Kiritchenko, Kiritchenko, Valentina
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1801.00334

Abstract

We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition (s₁)(s₂s₁)(s₃s₂s₁)(…)(s_n-1… s₁) of the longest element in the Weyl group, and the Schubert subvarieties correspond to the terminal subwords in this decomposition. We prove that the resulting Newton--Okounkov polytopes for semiample line bundles satisfy the additivity property with respect to the Minkowski sum. In particular, they are Minkowski sums of Newton--Okounkov polytopes of line bundles on the complete flag varieties for GL₂,…, GL_n.

Citations

• Bott towers, complete integrability, and the extended character of representations

Related

• Newton-Okounkov polytopes of flag varieties for classical groups
• Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules
• Singular string polytopes and functorial resolutions from Newton-Okounkov bodies
• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
• Newton-Okounkov bodies of flag varieties and combinatorial mutations