Research

My research is in the area of algebraic combinatorics, often among the intersection of total positivity, geometry of the flag variety, and representation theory of linear groups. Specifically, I am interested in

• Schubert Calculus and symmetric functions
• combinatorial representation theory
• cluster algebras, total positivity, and dimer models

Collaborators

Nick Ovenhouse (5), Gregg Musiker (4), Sunita Chepuri (3), Esther Banaian (2), Elizabeth Kelley (2), Ralf Schiffler (2), Calvin Yost-Wolff (2), Valerie Zhang (2), Michael Curran, Quang Dao, Claire Frechette, Greg Michel, Julian Wellman.

Papers Show Abstracts

Preprints

• Higer Dimer Covers on Snake Graphs. with G. Musiker, N. Ovenhouse, and R. Schiffler. (2023)
• Abstract. arXiv 2306.14389

• Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer covers (or perfect matchings) of snake graphs. Moreover, the enumeration of 1-dimer covers of snake graphs provides a combinatorial interpretation of continued fractions. In particular, the number of 1-dimer covers of the snake graph $\mathscr{G}[a_1,\ldots,a_n]$ is the numerator of the continued fraction $[a_1,\ldots,a_n]$. This number is equal to the top left entry of the matrix product $\left(\begin{smallmatrix} a_1&1\\1&0 \end{smallmatrix}\right) \cdots \left(\begin{smallmatrix} a_n&1\\1&0 \end{smallmatrix}\right)$. In this paper, we give enumerative results on $m$-dimer covers of snake graphs. We show that the number of $m$-dimer covers of the snake graph $\mathscr{G}[a_1,\ldots,a_n]$ is the top left entry of a product of analogous $(m+1)$-by-$(m+1)$ matrices. We discuss how our enumerative results are related to other known combinatorial formulas, and we suggest a generalization of continued fractions based on our methods. These generalized continued fractions provide some interesting open questions and a possibly novel approach towards Hermite's problem for cubic irrationals.

Published Articles.

• A Lattice Model for Super LLT Polynomials. with M. Curran, C. Frechette, C. Yost-Wolff, and V. Zhang. (2021)
• Abstract. arXiv 2110.07597 Comb. Theory 3(2). 2023.

• 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. Lastly, we construct a solvable semi-infinite Cauchy lattice model with a surprising Yang-Baxter equation and examine its connections to the Cauchy identity.

• Matrix Formulae for Decorated Super Teichmüller Spaces. with and . (2022)
• Abstract. arXiv 2208.13664 J. Geom. Phys. 2023, Vol.189.

• For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup OSp(1|2), which defines a flat 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 space. This generalizes the matrix formulas of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors’ previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers.

• Rooted Clusters for Graph LP Algebras. with E. Banaian, , and E. Kelley. (2021)
• Abstract. arXiv 2107.14785 SIGMA 18 (2022) 089

• 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.

• 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. Yost-Wolff. (2020)
• Abstract. arXiv 2002.04810 Electron. J. Comb. 29-2 (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 $K$-jeu-de-taquin 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 down-degree 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 deletion-contraction argument.

• An Expansion Formula for Decorated Super-Teichmüller Spaces. with and . (2021)
• Abstract. arXiv 2102.09143 SIGMA 17 (2021) 080

• Motivated by the definition of super Teichmüller spaces, and Penner-Zeitlin'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 super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$.  In particular, followinTg Penner-Zeitlin, 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.

Other Writings.

• Double Dimers and Super Ptolemy Relations. with G. Musiker and N. Ovenhouse. FPSAC Abstract. (2023)
• Rooted Clusters for Graph LP Algebras. with E. Banaian, , and E. Kelley. FPSAC Abstract. (2022)
• A Lattice Model for LLT Polynomials. with M. Curran, C. Yost-Wolff, and V. Zhang. REU Report. (2019)

You can also find my papers on arXiv, Google scholar and zbMATH.

Teaching

Current courses.

• I am not teaching in this school year.

Courses I've taught at the University of Minnesota.

• Math 4242 applied linear algebra.
• Math 2263 multivariable calculus.
• Math 1272 Calculus II.
• Math 1271 Calculus I.

Mentoring

PACE

I am a mentor at PACE –– PKU Algebraic Combinatorics Experience, and REU program at Peking University. Below is a list of projects I mentored.

[To be added.]

Twin Cities REU

• %-Immanants and Temperley-Lieb Immanants by Frank Lu, Kevin Ren, Dawei Shen, Siki Wang
• Scannable Divides of Finite Mutation Type by Pranjal Dangwal, Ryan Lynch, Son Nguyen and Ethan Pesikoff.
• Minimal Matchings for dP3 Cluster Variables by Judy Chiang and Son Nguyen.

Miscelleny

The content of this website is current as of January 2024.