Research – mathlab

I work mostly on combinatorics related to algebraic geometry and representation theory, though I also have an abiding interest in algorithmic number theory and computational complexity. Most of my papers are on symmetric spaces and their embeddings, which provide a nice context for generalizing Coxeter group combinatorics to certain kinds of involutions or “clans.” I also have work on Kazhdan-Lusztig polynomials, Kohnert polynomials, poset shellability, and elementary number theory. Recently, I am working on methods of computing Kronecker coefficients in the combinatorial representation theory of the symmetric group, as well as other collaborative projects involving chromatic symmetric functions, and algebraic coding theory. Below are links and descriptions for some of my papers.

Someday, works around other themes such as saturation, unimodality, functional digraphs, diagrammatic calculus (projects at varying stages of completion with diverse collaborators) will appear here too. If any of these topics interest you, feel free to get in touch.

Kohnert posets and polynomials of northeast diagrams, with Beth Anne Castellano, Kimberly P. Hadaway, Reuven Hodges, Yichen Ma, Alex Moon, Kyle Salois, 2025: This project came out of a problem Beth Anne proposed at GRWC 2024! We initially wanted to study the Kohnert posets of right-aligned “lock” (as opposed to “key”) diagrams. After making some fast progress at the workshop, we set our sights a little higher and enlarged our scope to a class of diagrams which are called “northeast.” We proved criteria for deciding when theses posets are ranked and bounded, as well as characterized monomial multiplicity-free Kohnert polynomials of northeast diagrams. Along the way, we collected some additional results on diagram enumeration and shellability of lock posets that will appear at some point soon. A subset of this group is continuing to work on the saturated Newton polytope property of Kohnert polynomials.

Lexicographic shellability of sects, with Néstor Díaz Morera, 2023: We show that the sects of the type $AIII$ symmetric variety are EL-shellable posets. We started this project attempting to show that the full Bruhat poset of Borel orbits on the symmetric variety (i.e. the poset of (p,q)-clans) was lexicographically shellable, but ran into some obstacles that we couldn’t see how to overcome, even using more flexible tools from the shellability toolkit like CL-labellings or CC-labellings. We still believe it to be the case that Bruhat posets on homogeneous spaces like symmetric varieties should be shellable. But we found it a nice consolation to use a bijection between sect orbits and rook placements in partition shapes, along with ideas from work of our adviser Mahir Can to obtain this result. It also surprised us that, somehow, this seems to be the time shellability of the poset of matrix Schubert varieties in rectangular matrices has been explicitly established.

Kronecker coefficients in the (p,q,2)-case via polytopes, with Ernesto Vallejo (coming soon): We recover and extend several results about Kronecker coefficients when one of the partitions has two parts using certain polytopes.

Ternary arithmetic, factorization, and the class number one problem, 2020: A commutative ternary “multiplication” operation on natural numbers can be defined by counting lattice points in the hexagonal lattice. It turns out that the “prime” numbers for this multiplication correspond to certain imaginary quadratic fields with unique factorization. I also describe some simple algorithms for primality testing and integer factorization. (It turns out that the factorization algorithm described is essentially the same as what is sometimes called “Kraitchik’s algorithm.”) This grew out of research I did for my master’s thesis. (Revista Colombiana de Matemáticas, 2022)

DIII clan combinatorics for the orthogonal Grassmannian, with Ozlem Ugurlu, 2019: We provide some bijective combinatorics for the parameter set of Borel orbits in a symmetric space of type $DIII$ $(SO_{2n}/GL_n)$ , which consists of clans with certain conditions. For clarity, we somewhat uncreatively called these “DIII clans,” though in retrospect I sort of wish we had gone with “spinor clans,” since part of the fun is once again identifying the sects corresponding to Schubert cells of the connected component of the even orthogonal Grassmannian which identifies with what is often called the spinor variety. Some other neat coincidences with rook placements, set partitions and weighted Delannoy paths show up as well. (Australasian Journal of Combinatorics, 2021)

Sects and lattice paths over the Lagrangian Grassmannian, with Ozlem Ugurlu, 2019: We enumerate and biject the clans parametrizing the $B$ -orbits classical symmetric space $Sp_{2n}/GL_n$ to certain restricted involutions and weighted Delannoy paths. We hoped to use the Delannoy paths to eventually describe the “Bruhat” or closure order on the clans, but this turned out to be unnecessary after discovering work of Gandini-Maffei. We were able to describe the sects and find a nice “big sect” coincidence with another Bruhat poset of partial involutions. (Electronic Journal of Combinatorics, 2020)

Sects, with Mahir Can, 2018: We describe local trivializations that give an affine bundle structure to the natural projection map $\pi: G/L \to G/P$ where $G$ is a reductive complex linear algebraic group and $P$ is a parabolic subgroup with Levi factor $L$ . This gives a cell decomposition of the affine variety $G/L$ which allows identification of its cohomology and Chow rings with those of $G/P$ . The details of the case $GL_{p+q}/GL_p \x GL_q\to \text{Gr}_p(\C^{p+q})$ are spelled out, combinatorially describing the collections of clans that constitute each cell, which we call a sect. (Journal of Algebra, 2020)

A filtration on equivariant Borel-Moore homology, with Mahir Can and Yildiray Ozan, 2018: We find some splittings of long exact sequences for relative (equivariant) Borel-Moore homology and Chow groups that allow us to obtain direct sum decompositions over the orbits for varieties with finitely many $G$ -orbits. In particular, this applies to wonderful embeddings of spherical homogeneous spaces $G/H$ including symmetric varieties. Together with results of the “Sects” paper, one has a complete description of the cohomology of symmetric spaces of Hermitian type and their embeddings. (Forum of Mathematics, Sigma, 2019).