Grant T. Barkley

Me Ph.D. Student
Department of Mathematics
Harvard University
Email: firstinital + lastname at
Pronouns: he, him, his

Hi! I am a fourth year mathematics Ph.D. student at Harvard University. My advisor is Lauren Williams. Before I was here, I was an undergraduate at NC State University. Here is my CV.

I'm interested in problems related to Lie and quiver representation theory, especially from combinatorial, geometric, and categorical viewpoints. My work has especially focused on infinite root systems and the geometry of flag varieties.


In the 2021-2022 school year, I co-organized the Trivial Notions graduate seminar. More information can be found here.

In Summer 2021, I taught a tutorial called "Quantum Mechanics for the Mathematically-Minded". You can find the course notes here.


Papers and preprints

  1. Shards for the affine symmetric group.
    We show that shards in the type-\(\widetilde{A}\) Coxeter arrangement biject with the complete join-irreducible elements of the type-\(\widetilde{A}\) extended weak order. This is a fact which is true for the weak order of a finite Coxeter group, but it is notable in the type-\(\widetilde{A}\) case because there are complete join-irreducibles which do not come from regions of the Coxeter arrangement. We also give a combinatorial parametrization of both shards and join-irreducibles via cyclic arc diagrams.
  2. On combinatorial invariance of parabolic Kazhdan–Lusztig polynomials (with C. Gaetz).
    We show that several conjectures on the combinatorial invariance of parabolic KL polynomials are in fact equivalent to the classical combinatorial invariance conjecture for KL polynomials.
  3. Bender–Knuth Billiards in Coxeter Groups (with C. Defant, E. Hodges, N. Kravitz, and M. Lee).
    We introduce non-invertible Bender–Knuth toggles on a Coxeter group, which generalize Bender–Knuth involutions on linear extensions of a poset. These form a dynamical system which in some cases can be realized as billiards in a Coxeter arrangement. The construction depends on a choice of convex set in the Coxeter group; we describe many cases where every trajectory is drawn into this convex set, including Coxeter groups that are finite, rank 3, right-angled, or of type \(\widetilde{A}\) or \(\widetilde{C}\). We call these groups futuristic; we also show that the remaining affine Coxeter groups are not futuristic (in fact, they are ancient).
  4. Affine extended weak order is a lattice (with D. Speyer).
    [ArXiv] [FPSAC]
    We prove that the extended weak order introduced by Matthew Dyer, which is the poset of biclosed sets in a positive root system, is a complete lattice for affine Weyl groups. This is a special case of a decades-old conjecture of Dyer that this lattice property is true for any Coxeter group. To prove it, we introduce the notion of a clean hyperplane arrangement, which is defined by the property that its regions can be computed using only codimension 2 data. We show that root poset order ideals in a finite or rank 3 untwisted affine root system are clean. We prove that Dyer's two main conjectures on biclosed sets reduce to finding sufficiently many clean subarrangements of rank 3 Coxeter arrangements.
  5. Combinatorial invariance for Kazhdan–Lusztig \(R\)-polynomials of elementary intervals (with C. Gaetz).
    [ArXiv] [FPSAC]
    An interval in Bruhat order is called simple if the roots connecting its minimal element to its atoms are linearly independent. We prove that if two Bruhat intervals in the symmetric group are isomorphic, and they are isomorphic to a simple interval, then their \(R\)-polynomials are equal. In particular, this holds for intervals isomorphic to a lower interval in Bruhat order, so this generalizes the result of Brenti that two isomorphic lower intervals in the symmetric group have the same \(R\)-polynomial. Our proof uses hypercube decompositions, which were introduced by Blundell, Buesing, Davies, Veličković, and Williamson and discovered with the assistance of techniques from machine learning.
  6. Combinatorial descriptions of biclosed sets in affine type (with D. Speyer). To appear in Combinatorial Theory.
    [ArXiv] [FPSAC]
    We give a parametrization of biclosed sets in an affine root system in terms of faces of the Coxeter fan (equivalently, the permutahedron) of its associated finite root system. In classical types, we give an explicit combinatorial description of biclosed sets as total orders of the integers, analogous to the description of affine Weyl groups as permutation groups.
  7. Channels, Billiards, and Perfect Matching 2-Divisibility (with R. Liu). Electronic Journal of Combinatorics, Volume 28, Issue 2, 2021.
    [ArXiv] [FPSAC] [Journal]
    We give a lower bound on the power of 2 dividing the number of perfect matchings of a planar graph using vertex sets called channels. We build many tools for computing the channels of various graphs. As a result, we recover many results from the literature and apply our technique to a conjecture of Pachter. In nice situations, we can identify channels with billiard trajectories in the graph, which lets us use the theory of arithmetic billiards to explain the power of 2 appearing in the number of domino tilings of a rectangle.