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Pavel Galashin

Publications and source records attributed to Pavel Galashin.

At least 19 recordsLinked to original sources
arXiv2026

Amplituhedra and origami, I: tree level

We establish a precise correspondence between points of the $m=4$ tree momentum amplituhedron and origami crease patterns. As an application, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ tree amplituhedron both in momentum space and in momentum-twistor space. As another application, we show that every nondegenerate weighted planar bipartite graph $Γ$ admits a t-embedding, i.e., an embedding of the planar dual of $Γ$ such that the sum of angles of white (equivalently, black) faces around each vertex is equal to $π$.

hep-th↗
arXiv2026

Amplituhedra and origami, II: loop level

Building on the recently discovered origami-amplituhedron correspondence, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space. Along the way, we develop two natural "$L$-punctured" extensions of the positive Grassmannian and relate them via T-duality.

hep-th↗
arXiv2025

Comparing cluster algebras on braid varieties

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

math.AG↗
arXiv2024

Braid variety cluster structures, I: 3D plabic graphs

We introduce $3$-dimensional generalizations of Postnikov's plabic graphs and use them to establish cluster structures for type $A$ braid varieties. Our results include known cluster structures on open positroid varieties and double Bruhat cells, and establish new cluster structures for type $A$ open Richardson varieties.

math.CO↗
arXiv2023

$P$-associahedra

For each poset $P$, we construct a polytope $A(P)$ called the $P$-associahedron. Similarly to the case of graph associahedra, the faces of $A(P)$ correspond to certain nested collections of subsets of $P$. The Stasheff associahedron is a compactification of the configuration space of $n$ points on a line, and we recover $A(P)$ as an analogous compactification of the space of order-preserving maps $P\to\mathbb{R}$. Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of $n$ points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases.

math.CO↗
arXiv2023

Monotone links in DAHA and EHA

We define monotone links on a torus, obtained as projections of curves in the plane whose coordinates are monotone increasing. Using the work of Morton-Samuelson, to each monotone link we associate elements in the double affine Hecke algebra and the elliptic Hall algebra. In the case of torus knots (when the curve is a straight line), we recover symmetric function operators appearing in the rational shuffle conjecture. We show that the class of monotone links viewed as links in $\mathbb R^3$ coincides with the class of Coxeter links, studied by Oblomkov-Rozansky in the setting of the flag Hilbert scheme. When the curve satisfies a convexity condition, we recover positroid links that we previously studied. In the convex case, we conjecture that the associated symmetric functions are Schur positive, extending a recent conjecture of Blasiak-Haiman-Morse-Pun-Seelinger, and we speculate on the relation to Khovanov-Rozansky homology. Our constructions satisfy a skein recurrence where the base case consists of piecewise almost linear curves. We show that convex piecewise almost linear curves give rise to algebraic links.

math.CO↗
arXiv2023

Positroids, knots, and $q,t$-Catalan numbers

We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $\mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.

math.CO↗
arXiv2023

Totally nonnegative critical varieties

We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety Crit$^{\ge0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of Crit$^{\ge0}_f$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety Crit$^{\ge0}_{k,n}$ is homeomorphic to the second hypersimplex $Δ_{2,n}$.

math.AG↗
arXiv2022

Move-reduced graphs on a torus

We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by Goncharov-Kenyon, and gives a toric analog of Postnikov's results on a disk.

math.CO↗
arXiv2022

The twist for Richardson varieties

We construct the twist automorphism of open Richardson varieties inside the flag variety of a complex semisimple algebraic group. We show that the twist map preserves totally positive parts, and prove a Chamber Ansatz formula for it. Our twist map generalizes the twist maps previously constructed by Berenstein-Fomin-Zelevinsky, Marsh-Scott, and Muller-Speyer. We use it to explain the relationship between the two conjectural cluster structures for Richardson varieties studied by Leclerc and by Ingermanson.

math.RT↗
arXiv2022

Plabic links, quivers, and skein relations

We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic links. For a subclass of permutations arising from concave curves, we also provide isotopies with the corresponding Coxeter links. Second, we associate a point count polynomial to an arbitrary locally acyclic quiver. We conjecture an equality between the top $a$-degree coefficient of the HOMFLY polynomial of a plabic graph link and the point count polynomial of its planar dual quiver. We prove this conjecture for leaf recurrent plabic graphs, which includes reduced plabic graphs and plabic fences as special cases.

math.CO↗
arXiv2022

Rational Noncrossing Coxeter-Catalan Combinatorics

We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.

math.CO↗
arXiv2021

Critical varieties in the Grassmannian

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of the positroid varieties in the Grassmannian. The combinatorics of positroid varieties is captured by the dimer model on a planar bipartite graph $G$, and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of $G$. This model is invariant under square/spider moves on $G$, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of $G$. This extends our recent results for the critical Ising model, and simultaneously also includes the case of critical electrical networks. We systematically develop the basic properties of critical varieties. In particular, we study their real and totally positive parts, the combinatorics of the associated strand diagrams, and introduce a shift map motivated by the connection to zonotopal tilings and scattering amplitudes.

math.CO↗
arXiv2021

A formula for boundary correlations of the critical Ising model

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical $Z$-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.

math-ph↗
arXiv2021

The totally nonnegative Grassmannian is a ball

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.

math.CO↗
arXiv2021

Positroid varieties and cluster algebras

We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and Serhiyenko--Sherman-Bennett--Williams.

math.CO↗
arXiv2021

Positroid Catalan numbers

Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky homology of Coxeter links.

math.CO↗