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

Abstract

We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $G{L}_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition $(s_1)(s_2{s}_1)(s_3{s}_2{s}_1)(\dots )(s_{n-1}\dots s_1)$ 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 $G{L}_2$,\ldots, $G{L}_n$.

9 pages, preliminary version, comments are welcome

References in corpus (1)

• On numerical Newton-Okounkov bodies and the existence of Minkowski bases