Related papers: Folding procedure for Newton-Okounkov polytopes of…
The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this paper, we study Newton-Okounkov…
A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and…
A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a…
A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body has various kinds of…
A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection…
For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell and is…
A Newton-Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations…
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…
A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their…
We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from…
It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is…
In this survey I summarize the constructions of toric degenerations obtained from valuations and Gr\"obner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton…
Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and show that these…
We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in…
Let \Delta be the Okounkov body of a divisor D on a projective variety X. We describe a geometric criterion for \Delta to be a lattice polytope, and show that in this situation X admits a flat degeneration to the corresponding toric…
We compute Okounkov bodies of projective complexity-one T-varieties with respect to two types of invariant flags. In particular, we show that the latter are rational polytopes. Moreover, using results of Dave Anderson and Nathan Ilten, we…
In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related as s first step towards developing a global framework. In focus are toric degenerations…
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes…
Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the…
The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations…