My research lies at the interface of commutative algebra, algebraic combinatorics, representation theory, and algebraic geometry. I utilize tools from topological, tropical, and algebraic combinatorics to investigate homological aspects of algebras with a group action or some other combinatorial structure. Recently I have been especially interested in combinatorial Koszul algebras, homological aspects of determinantal varieties, and interactions between commutative algebra and tropical combinatorics.
13. Standard Monomials for Positroid Varieties (with Shiliang Gao and Daoji Huang).
We give an explicit characterization of the standard monomials of positroid varieties with respect to the Hodge degeneration, and give a Grobner basis for positroid varieties. As an application, we show that promotion on rectangular-shaped semistandard tableaux gives a bijection between standard monomials of a positroid variety and its cyclic shifts.
12. Alexander Duals of Symmetric Simplicial Complexes and Stanley-Reisner Ideals (with Katie Bruegge, Martina Juhnke, Uwe Nagel, and Sasha Pevzner).
This project was started at the REACT workshop in March 2021.
Given an ascending chain of Sym-invariant squarefree monomial ideals {I_n}, we study the corresponding chain of Alexander duals {I^\vee_n}. Using a novel combinatorial tool, which we call avoidance up to symmetry, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of I^\vee_n is given by a polynomial in n for sufficiently large n. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in n eventually. The former result implies that the number of Sym-orbits of primary components of I_n grows polynomially in n for large n. As another application, we show that the number of i-dimensional faces of the associated Stanley-Reisner complexes of I_n is also given by a polynomial in n for large n.
11. The MatrixSchubert package for Macaulay2 (with Sean Grate, Daoji Huang, Patricia Klein, Adam LaClair, Yuyuan Luo, and Joseph McDonough). To appear in Journal for Software in Algebra and Geometry.
This paper resulted from the 2023 Macaulay2 workshop in Minneapolis (for which I was a co-organizer), where Patricia Klein and I led this project.
We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct, display, and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also has tools for quickly computing invariants of such varieties and verifying whether a union of matrix Schubert varieties is an ASM variety
10. Koszulity, supersolvability, and Stirling representations (with Vic Reiner and Sheila Sundaram). To appear in Annals of Representation Theory.
Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.
9. Root Polytopes, Tropical Types, and Toric Edge Ideals (with Anton Dochtermann and Ben Smith). Algebraic Combinatorics, 8.1 (2025): 59-99.
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincón, these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.
8. GL-equivariant resolutions over Veronese Rings (with Mike Perlman, Sasha Pevzner, Vic Reiner, and Keller VandeBogert), Journal of the London Mathematical Society, Vol. 109 (2024): e12848. https://doi.org/10.1112/jlms.12848
We construct explicit GL-equivariant minimal free resolutions of certain (truncations of) modules of relative invariants over Veronese subrings in arbitrary characteristic. The free modules in the resolution correspond to certain skew Schur modules corresponding to "ribbon" or "skew-hook" diagrams, and the differentials at each step are surprisingly uniform. We then utilize the uniformity of these resolutions to explicitly compute information Tor and Hom between these modules and prove that they also have rather simple descriptions in terms of ribbon skew-Schur modules.
7. Rees Algebras of Closed Determinantal Facet Ideals (with Kuei-Nuan Lin and Whitney Liske), Journal of Pure and Applied Algebra (2024).
Using SAGBI basis techniques, we find Grobner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.
6. Polarizations of Powers of Graded Maximal Ideals (with Gunnar Fløystad and Henning Lohne), Journal of Pure and Applied Algebra, Vol 226, Issue 5 (2022)
This paper was developed during my seven-month research visit to the University of Bergen in 2019. We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x_1,x_2,...,x_m)^n of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case and also in the degree two case the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We conjecture that any polarization of an artinian monomial ideal defines a simplicial ball.
5. Polarizations and Hook Partitions (with Keller VandeBogert), Journal of Pure and Applied Algebra (2022)
In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies--Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the "spanning tree condition" of Almousa--Fløystad--Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called "restricted powers" of the graded maximal ideal.
4. Linear Strands of Initial Ideals of Determinantal Facet Ideals (with Keller VandeBogert), Communications in Algebra, DOI: 10.1080/00927872.2021.2002885
We construct an explicit linear strand for the initial ideal with respect to any diagonal term order < of an arbitrary DFI. In particular, we show that if a pure simplicial complex has no 1-nonfaces, then the Betti numbers of the linear strand of its associated DFI and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most two maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to <) of any DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the complex of boxes, introduced by Nagel and Reiner.
3. Determinantal Facet Ideals for Smaller Minors (with Keller VandeBogert), Archiv der Mathematik (2022).
A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic matrix indexed by the facets of a simplicial complex. We consider the more general notion of an r-DFI, which is generated by a subset of r-minors of a generic matrix indexed by the facets of a simplicial complex for some 0<r<n+1. We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of Delta form a reduced Groebner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case r=n, lcm-closedness is necessary for being a Groebner basis. We also give conditions on the maximal cliques of Delta ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of r-DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.
2. The Virtual Resolutions Package for Macaulay2 (with Juliette Bruce, Michael Loper, and Mahrud Sayrafi), Journal for Software in Algebra and Geometry, Vol 10 (2020), pp. 51-60. [arXiv]
This paper resulted from my participation in the Macaulay2 workshop at University of Wisconsin-Madison in April 2018.
We develop the VirtualResolution package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in P1×P2, providing sources for interesting virtual resolutions.
Counting polynomials over finite fields with given root multiplicities (with Melanie Matchett Wood) Journal of Number Theory, 136C (2014), pp. 394-402.
This paper resulted from my participation in the "Undergraduate Research Scholars" Program at the University of Wisconsin-Madison.
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). We also prove an analogous result on configuration spaces in the Grothendieck ring of varieties, suggesting new homological stabilization conjectures for configuration spaces of the plane.