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

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 $(s_1)(s_2s_1)(s_3s_2s_1)(\ldots)(s_{n-1}\ldots s_1)$ 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 $GL_2$,\ldots, $GL_{n}$.