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Computing toric degenerations of flag varieties
We compute toric degenerations arising from the tropicalization of the full flag varieties \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) embedded in a product ofExpand
Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs
We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphsExpand
String cone and superpotential combinatorics for flag and Schubert varieties in type A
TLDR
It is proved that the two cones associated to each pseudoline arrangement are unimodularly equivalent and Caldero's toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory is realized. Expand
The degree of the Hilbert-Poincar\'e polynomial of PBW-graded modules
In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules ofExpand
Toric degenerations of cluster varieties and cluster duality
We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degenerationExpand
Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Grobner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$Expand
Birational sequences and the tropical Grassmannian
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$,Expand
Full-Rank Valuations and Toric Initial Ideals
Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$Expand
R T ] 2 8 M ar 2 01 9 Birational sequences and the tropical Grassmannian
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns’ birational sequences and explain how they give rise to points in trop(Gr(k,C)), Speyer-Sturmfels’ tropicalExpand
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