3 citations · 8 across the 14 of their papers we have counts for
19 papers
Tropical cluster varieties, phylogenetic trees, and generalized associahedra
Igor Makhlin
We explicitly describe the tropicalization of a type C cluster variety by identifying it with the space of axially symmetric phylogenetic trees. We also study the signed tropicaliz…
Tropical cluster varieties of type C
Igor Makhlin
We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign…
Regular subdivisions, bounds on initial ideals, and categorical limits
George Balla, Daniel Corey, Igor Makhlin +1
Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a ge…
Tropicalizing binary geometries
Shelby Cox, Igor Makhlin
The type A cluster configuration space, commonly known as , is the very affine part of the binary geometry associated with the associahedron. The tropicalization…
Poset polytopes and pipe dreams: toric degenerations and beyond
Ievgen Makedonskyi, Igor Makhlin
We demonstrate how pipe dreams can be applied to the theory of poset polytopes to produce toric degenerations of flag varieties. Specifically, we present such constructions for mar…
Poset polytopes and pipe dreams: types C and B
Ievgen Makedonskyi, Igor Makhlin
The first part of this paper concerns type C. We present new explicitly defined families of algebro-combinatorial structures of three kinds: combinatorial bases in representations,…
Chain-order polytopes: toric degenerations, Young tableaux and monomial bases
Igor Makhlin
Our first result realizes the toric variety of every marked chain-order polytope (MCOP) of the Gelfand--Tsetlin poset as an explicit Gröbner (sagbi) degeneration of the flag variet…
On reduced arc spaces of toric varieties
Ilya Dumanski, Evgeny Feigin, Ievgen Makedonskyi +1
An arc space of an affine cone over a projective toric variety is known to be non-reduced in general. It was demonstrated recently that the reduced scheme structure is worth studyi…
Relative poset polytopes and semitoric degenerations
Evgeny Feigin, Igor Makhlin
The two best studied toric degenerations of the flag variety are those given by the Gelfand--Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial…
Beyond the Sottile-Sturmfels degeneration of a semi-infinite Grassmannian
Evgeny Feigin, Igor Makhlin, Alexander Popkovich
We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only…
Semitoric degenerations of Hibi varieties and flag varieties
Evgeny Feigin, Igor Makhlin
We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varie…
Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties
Igor Makhlin
The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by se…
PBW degenerate Schubert varieties: Cartan components and counterexamples
Igor Makhlin
In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerat…
Gelfand--Tsetlin degenerations of representations and flag varieties
Igor Makhlin
Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to t…
Weighted PBW degenerations and tropical flag varieties
Xin Fang, Evgeny Feigin, Ghislain Fourier +1
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type . These degenerations are labeled by degree functi…
Vertices of FFLV polytopes
Evgeny Feigin, Igor Makhlin
FFLV polytopes describe monomial bases in irreducible representations of and . We study various sets of vertices of FFLV polytopes. First, we…
A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem
Boris Feigin, Igor Makhlin
We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type , i.e. corresponding to the affine Lie algebra $…
Weyl's Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes
Igor Makhlin
We exploit the idea that the character of an irreducible finite dimensional -module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope…
Characters of Feigin-Stoyanovsky subspaces and Brion's theorem
Igor Makhlin
We give an alternative proof of the main result of the paper http://arxiv.org/abs/math/0112104, the proof relies on Brion's theorem about convex polyhedra. The result itself can be…