Research

I work mostly on combinatorics related to algebraic geometry and representation theory, though I also have an abiding interest in algorithmic number theory. 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 and some elementary number theory. Most recently, I am working on methods of computing Kronecker coefficients in the combinatorial representation theory of the symmetric group. Below are links and descriptions for some of my papers.

  • Kazhdan-Lusztig polynomials for \tilde{B_2}, with Karina Batistelli and David Plaza, 2021: We calculate recursive and explicit formulas for the canonical basis of the Hecke algebra for an affine Weyl group of type \tilde{B}_2, as well as descriptions of lower intervals of the Bruhat order on this group. Combining these pieces of information, one can (more) easily extract Kazhdan-Lusztig polynomials h_{x,y} for pairs of elements in this group. These are the first such comprehensive calculations for a Coxeter group in which the W-graph is not locally finite. (submitted)
  • 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. This grew out of research I did for my master’s thesis. (to appear in Revista Colombiana de Matem├íticas, 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).