UMN Combinatorics Seminar: Fall 2025

Title: Random Quotients of Free Groups

Abstract: The study of random groups is one way to answer the question, "What behavior is `typical' for groups?" In this talk I'll introduce several models of random groups, share what is currently known about groups using these models. and talk about recent work pushing our knowledge in the field. This work sits at the intersection of combinatorial and geometric group theory.

Title: Odd shifted parking functions

Abstract: Recently, Stanley introduced the shifted parking function symmetric function SH_n as the shiftification of the classical parking function symmetric function PF_n. In this talk, we build two actions of the symmetric group whose character is SH_n and a combinatorial map between them. The first is on signed parking functions, also introduced by Stanley, while the second is on a novel family of combinatorial objects we call odd shifted parking functions. These actions allow us to realize combinatorially Stanley's expansion of SH_n into the V-basis, analogous to the homogenous expansion of PF_n. We conclude with some open problems and future directions.

This is joint work with Jesse Kim.

Title: Asymptotics of algebraic generating functions

Abstract: Extracting asymptotics from a generating function is well-understood when the function is univariate or multivariate and rational.  But, a unified approach is lacking for more general multivariate generating functions.  Multivariate algebraic generating functions are an attractive next step: they can be used to enumerate Catalan objects, lattice paths in restricted domains, the outputs of RNA structure prediction methods, trees, and probability distributions.  In this talk, we will look at how to relate multivariate algebraic generating functions to their rational counterparts, making partial progress towards automating the procedure for this broader class.

Joint work with Stephen Melczer, Tiadora Ruza, and Mark C. Wilson.

Title: Hadamard Products of Dual Jacobi-Trudi Matrices

Abstract: In general, the Hadamard (i.e. entry-wise) product of matrices does not preserve the positivity of the determinant. A theorem of Wagner, however, states that positivity will be preserved for certain classes of matrices. Upgrading Wagner’s result to the setting of symmetric functions, Sokal conjectured that Hadamard products of Jacobi-Trudi matrices have monomial-positive determinants. In the present work, we investigate both Sokal’s original conjecture and a generalization involving Temperley-Lieb immanants.

Title: Rotationally symmetric plabic graphs and the Lagrangian Grassmannian

Abstract: In this talk, we introduce the totally nonnegative Lagrangian Grassmannian $\rm{LG}^R_{\geq 0}$, a new subset of the totally nonnegative Grassmannian $\rm{Gr}_{\geq 0}$, determined by a specific skew-symmetric bilinear form $R$. We describe its cell decomposition and show that each cell can be represented by a rotationally symmetric plabic graph. These graphs are not always reduced, which requires the development of new tools for working with non-reduced plabic graphs.

Title: Euler characteristics of K-classes for pairs of matroids 

Abstract: In his 2005 PhD thesis on tropical linear spaces, Speyer conjectured an upper bound on the number of interior faces in a matroid base polytope subdivision of a hypersimplex. This conjecture can be reduced to determining the sign of the Euler characteristic of a certain matroid class in the K-theory of the permutohedral variety. In a recent joint work with Alex Fink, we prove Speyer's conjecture by showing that the requisite Euler characteristic is non-positive for all matroids, and extend this to a statement about pairs of matroids on the same ground set. In this talk, I will provide an overview of our strategy and zoom in on how we extend geometric results for realizable pairs of matroids to all pairs.

Title: Lower Bound on Tree Covers

Abstract: Given an n-point metric space (X,dX), a tree cover T is a set of |T|=k trees on X such that every pair of vertices in X has a low-distortion path in one of the trees in T. Tree covers have been playing a crucial role in graph algorithms for decades, and the research focus is the construction of tree covers with small size k and distortion.

When k=1, the best distortion is known to be \Theta(n). For a constant k≥2, the best distortion upper bound is \tilde O(n^{1/k}) and the strongest lower bound is \Omega(log_k n), leaving a gap to be closed. In this paper, we improve the lower bound to \Omega(n^{1/2^{k−1}}).

Our proof is a novel analysis on a structurally simple grid-like graph, which utilizes some combinatorial fixed-point theorems. We believe that they will prove useful for analyzing other tree-like data structures as well.

This talk is based on a joint work with Yu Chen and Hangyu Xu.

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Title: The HOMFLY Polynomial of a Forest Quiver 

Abstract: The HOMFLY polynomial of a link is a two-variable link invariant which was introduced in the 1980s. It can be defined recursively using a skein relation and specializes to other link invariants such as the Alexander polynomial and Jones polynomial. We will study this polynomial for plabic links, a class of links which are associated to plabic graphs. In particular, for a plabic graph whose quiver is an orientation of a forest, we will describe how to compute the HOMFLY polynomial recursively in terms of the quiver and provide a closed form expression for it. 

Title: A descent basis for $R_{n,\lambda,s}$

Abstract: The $\Delta-$Springer modules $R_{n,\lambda,s}$,  defined by Griffin, is a generalization of the Type A coinvariant ring. These rings unify the stories of the generalized coinvariant rings $R_{n,k}$, introduced by Haglund—Rhoades—Shimozono, and the Garsia—Procesi rings $R_\lambda$.

We give a monomial basis of $R_{n,lambda,s}$ consisting of generalizations of Garsia—Stanton descent monomials, which simultaneously generalizes the descent bases of $R_{n,k}$ and $R_{\lambda}$. We highlight the representation theoretic properties of this monomial basis by using it to give a direct combinatorial proof of the graded Frobenius character of $R\nls$ in terms of battery-powered tableaux, a fact which has only the geometric proof of Gillespie-Griffin (2024). This is joint work with R. Chou.

Title: Permutohedral and Multipermutohedral Chow rings

Abstract: This talk will compare classical and recent results about the permutohedral Chow ring to results for a new object called the multipermutohedral Chow ring. This ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of  permutohedral varieties, type-B Coxeter arrangements, and delta matroids. We will discuss the combinatorial and representation theoretic structure of the permutohedral Chow ring to work toward a closed formula for the representation of this ring (due Liao) and equivariant unimodality and palindromicity, due to Angarone, Nathanson, and Reiner. We then discuss these results in the multipermutohedral setting.