M. Sherman-Bennett

activity

20172026

most citedCluster structures in Schubert varieties in the Grassmannian

18 citations · 50 across the 23 of their papers we have counts for

collaborators

24 papers

math.AG2026

Permutahedra, Lusztig varieties, degenerations, and subdivisions

Allen Knutson, Mario Sanchez, Melissa Sherman-Bennett

We present an embedded (in $G / B$ ) degeneration of Lusztig varieties (which generalize type $A$ Hessenberg varieties) to certain unions of Richardson varieties, giving a simultaneou…

math.CO2026

Persistent Subdivisions of Coxeter Permutahedra

Timothy Blanton, Jesús A. De Loera, Melissa Sherman-Bennett

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $conv (W \cdot a)$ where $W$ is a…

math.AG2026

Unexpected toric Richardson varieties

Eugene Gorsky, Soyeon Kim, Melissa Sherman-Bennett

We prove that an open Richardson variety in the complete flag variety for $GL_{n}$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is tor…

math.CO2025

A dimer view on Fox's trapezoidal conjecture

Karola Mészáros, Melissa Sherman-Bennett, Alexander Vidinas

Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remain…

math.CO2025

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

Colin Defant, Melissa Sherman-Bennett, Nathan Williams

For each finite Coxeter group $W$ and each standard Coxeter element of $W$ , we construct a triangulation of the $W$ -permutahedron. For particular realizations of the $W$ -permutahed…

math.AG2025

Comparing cluster algebras on braid varieties

Roger Casals, Pavel Galashin, Mikhail Gorsky +3

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, posi…

math.CO2025

Plabic Tangles and Cluster Promotion Maps

Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett +2

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between…

math.CO2025★ 3 cited

BCFW tilings and cluster adjacency for the amplituhedron

Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi +3

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and…

math.CO2024

Dimer face polynomials in knot theory and cluster algebras

Karola Mészáros, Gregg Musiker, Melissa Sherman-Bennett +1

The set of perfect matchings of a connected bipartite plane graph $G$ has the structure of a distributive lattice, as shown by Propp, where the partial order is induced by the heig…

math.CO2024

The Magic Number Conjecture for the $m = 2$ amplituhedron and Parke-Taylor identities

Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler +1

The amplituhedron $A_{n, k, m}$ is a geometric object introduced in the context of scattering amplitudes in $N = 4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=…

math.CO2024★ 7 cited

A cluster of results on amplituhedron tiles

Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi +3

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $N = 4$ super Yang Mills theory. It generalizes \e…

math.CO2023★ 1 cited

Cluster algebras and tilings for the m=4 amplituhedron

Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi +3

The amplituhedron $A_{n, k, m} (Z)$ is the image of the positive Grassmannian $G r_{k, n}^{\geq 0}$ under the map $Z : G r_{k, n}^{\geq 0} \to G r_{k, k + m}$ induced by a positive linear ma…

math.CO2023★ 2 cited

Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

Roger Casals, Ian Le, Melissa Sherman-Bennett +1

First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from…

math.AG2023★ 3 cited

Braid variety cluster structures, II: general type

Pavel Galashin, Thomas Lam, Melissa Sherman-Bennett

We show that braid varieties for any complex simple algebraic group $G$ are cluster varieties. This includes open Richardson varieties inside the flag variety $G / B$ .

math.CO2022

Leclerc's conjecture on a cluster structure for type A Richardson varieties

Khrystyna Serhiyenko, Melissa Sherman-Bennett

Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structur…

math.CO2022★ 3 cited

Braid variety cluster structures, I: 3D plabic graphs

Pavel Galashin, Thomas Lam, Melissa Sherman-Bennett +1

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 clust…

math.CO2021

$k$ -positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan-Lusztig immanants

Sunita Chepuri, Melissa Sherman-Bennett

In this note, we show that certain dual canonical basis elements of $C [S L_{m}]$ are positive when evaluated on $k$ -positive matrices, matrices whose minors of size $k \times…

math.CO2021★ 9 cited

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

Matteo Parisi, Melissa Sherman-Bennett, Lauren Williams

The hypersimplex $Δ_{k + 1, n}$ is the image of the positive Grassmannian $G r_{k + 1, n}^{\geq 0}$ under the moment map. It is a polytope of dimension $n - 1$ in $R^{n}$ . Meanwhile,…

math.CO2020

Saturation of Newton polytopes of type A and D cluster variables

Amal Mattoo, Melissa Sherman-Bennett

We study Newton polytopes for cluster variables in cluster algebras $A (Σ)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster…

math.CO2020★ 1 cited

1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity

Sunita Chepuri, Melissa Sherman-Bennett

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve $q = 1$ specializations o…

math.CO2019★ 18 cited

Cluster structures in Schubert varieties in the Grassmannian

K. Serhiyenko, M. Sherman-Bennett, L. Williams

In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is en…

math.CO2018★ 1 cited

Combinatorics of cluster structures in Schubert varieties

Khrystyna Serhiyenko, Melissa Sherman-Bennett, Lauren Williams

We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov's plabic graphs. This desc…

math.CO2018

Combinatorics of $X$ -variables in finite type cluster algebras

Melissa Sherman-Bennett

We compute the number of $X$ -variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classi…

math.CO2017★ 2 cited

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Glencora Borradaile, Hung Le, Melissa Sherman-Bennett

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has a…

Permutahedra, Lusztig varieties, degenerations, and subdivisions

Persistent Subdivisions of Coxeter Permutahedra

Unexpected toric Richardson varieties

A dimer view on Fox's trapezoidal conjecture

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

Comparing cluster algebras on braid varieties

Plabic Tangles and Cluster Promotion Maps

BCFW tilings and cluster adjacency for the amplituhedron

Dimer face polynomials in knot theory and cluster algebras

The Magic Number Conjecture for the m=2 amplituhedron and Parke-Taylor identities

A cluster of results on amplituhedron tiles

Cluster algebras and tilings for the m=4 amplituhedron

Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

Braid variety cluster structures, II: general type

Leclerc's conjecture on a cluster structure for type A Richardson varieties

Braid variety cluster structures, I: 3D plabic graphs

k-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan-Lusztig immanants

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

Saturation of Newton polytopes of type A and D cluster variables

1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity

Cluster structures in Schubert varieties in the Grassmannian

Combinatorics of cluster structures in Schubert varieties

Combinatorics of X-variables in finite type cluster algebras

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

The Magic Number Conjecture for the $m = 2$ amplituhedron and Parke-Taylor identities

$k$ -positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan-Lusztig immanants

Combinatorics of $X$ -variables in finite type cluster algebras