# Sylvester ZhangUniversity of Minnesota Twin Cities | UMN · Department of Mathematics

Sylvester Zhang

Bachelor of Science

## About

12

Publications

260

Reads

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14

Citations

Citations since 2016

Introduction

I work on algebraic combinatorics, topics that I'm currently thinking about include cluster and LP algebras, tableaux and rep theory, and some discrete dynamical systems.

**Skills and Expertise**

Additional affiliations

September 2019 - present

Education

September 2016 - May 2020

## Publications

Publications (12)

Motivated by the definition of super-Teichmüller spaces, and Penner–Zeitlin’srecent extension of this definition to decorated super-Teichm ̈uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for superλ-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able...

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super $\lambda$-lengths in a marked disk, generalizing Schiffler's $T$-path formula. In the present paper, we give an alternate combinatorial expression for these super $\lambda$-lengths in terms of double dimer covers on snake graphs. This generalizes the dime...

An arborescence of a directed graph $\Gamma$ is a spanning tree directed toward a particular vertex $v$. The arborescences of a graph rooted at a particular vertex may be encoded as a polynomial $A_v(\Gamma)$ representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph $\tilde...

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove...

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove...

For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup $\mathrm{OSp}(1|2)$, which defines a flat $\mathrm{OSp}(1|2)$-connection on the surface. We show that our matrix formulas of an arc yields its super $\lambda$-length in Penner-Zeitlin's decorated super Teichm\"uller spac...

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler's T-path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musi...

Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a...

We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super n-ribbon tableaux. Using operators on a Fock space, we prove a Cauchy identity for super LLT polynomials, simultaneously generalizing the Cauchy and dual Cauchy identities for LLT polynomials. Last...

Motivated by the definition of super Teichm\"{u}ller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super Teichm\"{u}ller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a di...

Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a...

## Projects

Project (1)