arXiv subjects

Melissa Sherman-Bennett

Publications and source records attributed to Melissa Sherman-Bennett.

28 records · Page 2Linked to original sources

arXiv2023

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

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 a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.

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arXiv2022

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

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 structure is in fact a cluster structure. We do this by comparing Leclerc's construction with another cluster structure on type A Richardson varieties due to Ingermanson. Ingermanson's construction uses the combinatorics of wiring diagrams and the Deodhar stratification. Though the two cluster structures are defined very differently, we show that the quivers coincide and clusters are related by the twist map for Richardson varieties, recently defined by Galashin--Lam.

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arXiv2022

Positroid cluster structures from relabeled plabic graphs

The Grassmannian is a disjoint union of open positroid varieties $P_v$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring of $P_v$ is a cluster algebra, and each reduced plabic graph $G$ for $P_v$ determines a cluster. We study the effect of relabeling the boundary vertices of $G$ by a permutation $r$. Under suitable hypotheses on the permutation, we show that the relabeled graph $G^r$ determines a cluster for a different open positroid variety $P_w$. As a key step of the proof, we show that $P_v$ and $P_w$ are isomorphic by a nontrivial twist isomorphism. Our constructions yield many cluster structures on each open positroid variety $P_w$, given by plabic graphs with appropriately relabeled boundary. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and Laurent monomial transformations involving frozen variables, and establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs $G$, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.

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arXiv2021

Saturation of Newton polytopes of type A and D cluster variables

We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(Σ)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed $Σ$. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if $Σ$ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are \emph{empty}; that is, when they have no non-vertex lattice points.

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arXiv2021

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

In this note, we show that certain dual canonical basis elements of $\mathbb{C}[SL_m]$ are positive when evaluated on $k$-positive matrices, matrices whose minors of size $k \times k$ and smaller are positive. Skandera showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be written in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan-Lusztig immanants and uses similar tools, namely Lewis Carroll's identity (also known as the Desnanot-Jacobi identity).

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arXiv2021

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

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan-Lusztig polynomials, and were defined in (Rhoades-Skandera, 2006). Using results of (Haiman, 1993) and (Stembridge, 1991), Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on $k$-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). We show that the Kazhdan-Lusztig immanant indexed by $v$ is positive on $k$-positive matrices when $v$ avoids 1324 and 2143 and for all non-inversions $i<j$ of $v$, either $j-i \leq k$ or $v_j-v_i\leq k$. Our main tool is Lewis Carroll's identity.

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arXiv2019

Combinatorics of $\mathcal{X}$-variables in finite type cluster algebras

We compute the number of $\mathcal{X}$-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classical types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.

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arXiv2018

Combinatorics of cluster structures in Schubert varieties

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 description is a natural generalization of the description given by (Scott 2006) for the Grassmannian and has been believed by experts essentially since (Scott 2006), though the statement was not formally written down until (Müller-Speyer 2016). To prove this conjecture we use a result of (Leclerc 2016), who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety admit cluster structures. We also adapt a construction of (Karpman 2016) to build cluster seeds associated to reduced expressions. Further, we explicitly describe cluster structures in skew Schubert varieties using plabic graphs whose boundary vertices need not be labeled in cyclic order.

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arXiv2015

Talk to the Hand: Generating a 3D Print from Photographs

This manuscript presents a linear algebra-based technique that only requires two unique photographs from a digital camera to mathematically construct a 3D surface representation which can then be 3D printed. Basic computer vision theory and manufacturing principles are also briefly discussed.

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arXiv2013

3D Printing for Math Professors and Their Students

In this primer, we will describe a number of projects that can be completed with a 3D printer, particularly by mathematics professors and their students. For many of the projects, we will utilize Mathematica to design objects that mathematicians may be interested in printing. Included in the projects that are described is a method to acquire data from an XBox Kinect.

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