My Research
My research interest is in algebraic combinatorics. Specifically, I am interested in combinatorial representation theory, cluster algebras, discrete dynamical systems, and integrable models.
Publications Show Abstracts
Published Articles.
 Double Dimer Covers on Snake Graphs from Super Cluster Expansions. with and . (2021)
 Abstract. arXiv 2110.06497 J. Algebra Vol 608 (2022)

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 dimer formulas of Musiker, Schiffler, and Williams.
 Rowmotion Orbits of Trapezoid Posets. with Q. Dao, J. Wellman and C. YostWolff. (2020)
 Abstract. arXiv 2002.04810 Electron. J. Comb. 292 (2022)

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 conjecture of Hopkins that the rectangle and trapezoid posets have the same rowmotion orbit structures. Our main tools in proving this are $K$jeudetaquin and (weak) $K$Knuth equivalence of increasing tableaux. We define almost minimal tableaux as a family of tableaux naturally arising from order ideals and show for any $\lambda$, the almost minimal tableaux of shape $\lambda$ are in different (weak) $K$Knuth equivalence classes. We also discuss and make some progress on related conjectures of Hopkins on downdegree homomesy.
 Arborescences of Covering Graphs. with , CJ Dowd, A. Hardt, G. Michel, and V. Zhang. (2019)
 Abstract. arXiv 1912.01060 Algebr. Comb. Vol. 5 (2022)

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 $\tilde{\Gamma}$ are closely related. Using voltage graphs to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of $A_v(\Gamma)$ to the sum of arborescences in the lift $A_{\tilde{v}}(\tilde{\Gamma})$ in terms of the determinant of Chaiken's voltage Laplacian matrix, a generalization of the Laplacian matrix. Chaiken's results on the relationship between the voltage Laplacian and vector fields on $\Gamma$ are reviewed, and we provide a new proof of Chaiken's results via a deletioncontraction argument.
 An Expansion Formula for Decorated SuperTeichmüller Spaces. with and . (2021)
 Abstract. arXiv 2102.09143 SIGMA 17 (2021) 080

Motivated by the definition of super Teichmüller spaces, and PennerZeitlin's recent extension of this definition to decorated super Teichmüller 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 borderded surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$path formulas for type $A$ cluster algebras. We further connect our formulas to the superfriezes of MorierGenoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, followinTg PennerZeitlin, we are able to get formulas (up to signs) for the $\mu$invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.
Preprints and Submitted Papers.
 A Lattice Model for Super LLT Polynomials. with M. Curran, C. Frechette, C. YostWolff, and V. Zhang. (2021)
 Abstract. arXiv 2110.07597

We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super nribbon 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. Lastly, we construct a solvable semiinfinite Cauchy lattice model with a surprising YangBaxter equation and examine its connections to the Cauchy identity.
 Rooted Clusters for Graph LP Algebras. with E. Banaian, , and E. Kelley. (2021)
 Abstract. arXiv 2107.14785

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 positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of $T$paths.
Unpublished Writings.
 A Lattice Model for LLT Polynomials. with M. Curran, C. YostWolff, and V. Zhang. (2019)
 Abstract. Preprint (PDF)

We construct a family of $2$ dimensional lattice models depending on a positive integer $n$ whose partition functions are equal to the LLT polynomials of Lascoux, Leclerc and Thibon (originally named $n$ribbon Schur functions). We conjecture that our lattice model is solvable for all $n$, and compute the YangBaxter equations for up to $n=3$.
You can also find my papers on my arXiv author page, Google scholar profile and zbMATH.
Talks
 Combinatorial Formulas for Graph LP Algebras. UMN Student Combinatorics Seminar. (April 2022)
 Schur and LLT Polynomials from Lattice Models. Graduate Online Combinatorics Colloquium. (Mar 2021)
 $T$path Formula for Decorated SuperTeichmüller Spaces. UMN Combinatorics Seminar. (Feb 2021)
 Decorated SuperTeichmüller Spaces and Cluster Superalgebras. UMN Student Combinatorics Seminar. (Jan 2020)
Events
 The LA Workshop on Representation Theory and Geometry. USC, LA (June 2022)
 Minnesota Research Workshop in Algebra and Combinatorics, UMN, Minneapolis (May 2022)
 Open Problems in Algebraic Combinatorics, UMN, Minneapolis (May 2022)
 Great Plains Combinatorics Conference, NDSU, North Dakota (April 2022)
 Workshop on Supergeometry and Bracket Structures, Fields Institute, Toronto (March 2022)
 Dynamical Algebraic Combinatorics, BIRS, virtual (Nov 2021)
Codes
 Arborescences of covering graphs SageMath 9
 LLT polynomials Python 3 + SymPy
 Birational Rowmotion Python 3 + SymPy
 KenyonWilson Groves SageMath 9
 Graph LP Algebras Wolfram Mathematica
Miscellany
Combinatorics at Minnesota
© Wenze Zhang. All rights reserved.
The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.