Singular string polytopes and functorial resolutions from Newton-Okounkov bodies

2017/12/28 by Harada, Megumi, Yang, Jihyeon Jessie
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1712.09788

Abstract

The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation ν_max coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson varieties with respect to a different valuation ν_min in terms of Grossberg-Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton-Okounkov bodies coincide.

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