Illinois Journal of Mathematics

Singular string polytopes and functorial resolutions from Newton–Okounkov bodies

Megumi Harada and Jihyeon Jessie Yang

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The main result of this paper is that the toric degenerations of flag and Schubert varieties associated to string polytopes and certain Bott–Samelson resolutions of flag and Schubert 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 $\nu_{\mathrm{max}}$ 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 $\nu_{\mathrm{min}}$ in terms of Grossberg–Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton–Okounkov bodies coincide.

Article information

Illinois J. Math., Volume 62, Number 1-4 (2018), 271-292.

Received: 21 November 2018
Revised: 21 November 2018
First available in Project Euclid: 13 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
Secondary: 20G05: Representation theory


Harada, Megumi; Yang, Jihyeon Jessie. Singular string polytopes and functorial resolutions from Newton–Okounkov bodies. Illinois J. Math. 62 (2018), no. 1-4, 271--292. doi:10.1215/ijm/1552442663.

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