UMN Combinatorics Seminar: Spring 2026

Title: Linear quotients of powers of edge ideals

Abstract: The edge ideal $I(G)$ of a finite simple graph $G$ is the squarefree quadratic monomial ideal generated by monomials corresponding to the edges of $G$. Nevo and Peeva conjectured that if $G$ is gap-free, then $I(G)^s$ has a linear resolution for all sufficiently large $s$. In this talk, we explore the linear quotients property and discuss recent progress on this conjecture.

Joint work with S. Faridi, T. Ha, T. Hibi, S. Kara and S. Morey.

Title: A positive combinatorial formula for the double Edelman-Greene coefficients

Abstract: Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the associated double Edelman--Greene coefficients, the double Schur expansion coefficients of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. In this paper, we provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains. This is based on joint work with Tianyi Yu. 

Title: Invariant theory and wreath products

Abstract: Invariant theory starts with a group G acting on a polynomial algebra via linear substitutions of the variables, and tries to describe the subalgebra of G-invariant polynomials.  The theory is particularly well-developed when the group G is finite and the polynomial algebra has coefficients in a field of characteristic zero.

After reviewing this, we will explain how certain data (Hilbert series) for the G-invariant polynomials determines the same data for the S_n[G]-invariant polynomials for all n.  Here S_n[G] is the wreath product group G wr S_n acting on n sets of the original variables, with the symmetric group S_n swapping the variable sets.  This then suggests a structural result:  one can collate the direct sum on n of the S_n[G]-invariants into one big graded ring with a shuffle product, and this big ring ends up being a "superpolynomial algebra" (=symmetric tensor exterior algebra) generated by the G-invariants.  

(Based on arXiv:2503.19323 with Trevor Karn).

Title: Matrix loci and shadow play

Abstract: The {\em RSK correspondence} gives a bijection from permutations in $\mathfrak{S}_n$ and ordered pairs $(P,Q)$ of $n$-box standard Young tableaux of the same shape. Viennot gave a beautiful geometric reformulation of this bijection with a `shadow line' construction. We give an algebraic perspective on Viennot's construction via a quotient of the polynomial ring over an $n \times n$ matrix $x = (x_{i,j})$ of variables. This quotient ring arises from the {\em orbit harmonics} technique of combinatorial deformation theory, and opens the door to a family of connections between the combinatorics of matrix loci and the algebra of their associated graded rings. Joint with Jasper Liu, Yichen Ma, Jaeseong Oh, and Hai Zhu.

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Title: Line Shellings of Geometric Lattices

Abstract: Inspired by Bruggesser and Mani’s line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given any nested set complex of a matroid, we show that every lexicographic ordering of the vertices of the corresponding normal complex induces a shelling order. This gives a new proof of Björner’s classical result that the order complex of the lattice of flats of a matroid is shellable, and demonstrates shellability for all nested set complexes of matroids. This talk is based on joint work with Spencer Backman, Galen Dorpalen-Barry, Anastasia Nathanson and Noah Prime.

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Title: Kirillov-Reshetikhin dual equivalence graphs

Abstract: We introduce Kirillov-Reshetikhin dual equivalence graphs, a generalization of Assaf's dual equivalence graphs to affine type A. Given a sequence of rectangular partitions, we use commutators of adjacent crystal operators to construct a graph on the 0-weight space of the corresponding KR crystal. We give a complete characterization of the set of edges that appear in these graphs, which includes standard dual equivalence moves as a subset. Furthermore, we characterize the number of connected components in terms of the energy function of the underlying crystal. We also conjecture a plethysm formula for characters of these connected components involving cyclic characters. This talk is based on joint work with Pasha Pylyavskyy and Shiyun Wang.

Title: Open quiver loci, CSM classes, and chained generic pipe dreams

Abstract: In the space of type A quiver representations, putting rank conditions on the maps cuts out subvarieties called "open quiver loci." These subvarieties are closed under the group action that changes bases in the vector spaces, so their closures define classes in equivariant cohomology, called "quiver polynomials." Knutson, Miller, and Shimozono found a pipe dream formula to compute these polynomials in 2006. To study the geometry of the open quiver loci themselves, we might instead compute "equivariant Chern-Schwartz-MacPherson classes," which interpolate between cohomology classes and Euler characteristic. I will introduce objects called "chained generic pipe dreams" that allow us to compute these CSM classes combinatorially, and along the way give streamlined formulas for quiver polynomials.

Title: Zamolodchikov Periodic Cluster Algebras 

Abstract: Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz for simply-laced Dynkin diagrams, and was proved by Keller to hold for tensor products of two Dynkin diagrams. In this talk, we discuss the classification of all Zamolodchikov periodic cluster algebras, with connections to W-graphs, root systems, and maximal green sequences.

Title: Webs, pockets, and buildings

Abstract: Kuperberg’s SL(3) non-elliptic web basis consists of certain trivalent planar graphs. Fontaine--Kamnitzer--Kuperberg showed that their duals may be realized as subcomplexes of a corresponding rank 2 affine building. The result is a collection of CAT(0) triangulated surfaces related to the geometric Satake correspondence. Recently, an SL(4) web basis was introduced by Gaetz--Pechenik--Pfannerer--Striker--S. which comes with "moves". We show the moves may be understood geometrically as forming "pockets", certain highly structured 3D simplicial subcomplexes of the corresponding rank 3 affine building. These pockets have extraordinarily rich combinatorial structure. Special cases correspond to plane partitions, alternating sign matrices, tilings of the Aztec diamond, and more. Joint with Christian Gaetz, Jessica Striker, and Haihan Wu.

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