My research lies in the intersection of combinatorics and representation theory. I am interested in better understanding algebraic structures by modelling them through combinatorial structures with simple diagrammatic rules.
A complete list of my publications can be found on the arXiv. Some selected ones are described below.
Rotation invariant webs for flamingo Specht modules (2024+)
Webs are planar networks which give a diagrammatic way of understanding the representation category of the quantum group of SLr. When r=2,3,4, webs can be used to give a basis of the irreducible Sn modules corresponding to r-row rectangular partitions for which the action of the long cycle is simply diagram rotation. We show how to extend webs to give a similar basis for irreducible Sn modules corresponding to not-quite-rectangular partitions called flamingo shapes, so called because they appear to stand on one leg. As a result, we obtain a cyclic sieving result for the q-hook length formula for these partition shapes. This work generalizes the skein action on set partitions, which we recover when r=2.
A pentagonal number theorem for tribone tilings with Jim Propp (2023)
Conway and Lagarias showed that certain triangular regions of the hex grid could not be tiled by “tribones,” tiles consting of three adjacent hexes in a line, by utilizing Cayley graphs to develop an invariant. We study an extension of this to a two-parameter family of roughly hexagonal regions in the hex grid and show that such regions can be tiled if and only if the parameters are paired pentagonal numbers.
Combinatorics of supertorus sheaf cohomology with Brendon Rhoades and Jeff Rabin (2023)
Superspace is a space arising in mathematical physics to describe the state space of a collection of a combination of bosons and fermions. We study the cohomology of the structure sheaf of a generalization of the torus within this space. We give a combinatorial model of the 0th and 1st cohomolgy in terms of arc diagrams and use this model to extract a linear basis.
Set partitions, fermions, skein relations with Brendon Rhoades (2023)
The symmetric groups Sn acts diagonally on an exterior algebra on two sets of n variables. The fermionic diagonal coinvariant ring is the quotient of this exterior algebra by the ideal generated by Sn invariants with vanishing constant term. We give an Sn equivariant isomorphism between the top degree piece of this ring and the skein action on formal linear combinations of set partitions and use this to compute the Frobenius image of the fermionic diagonal coinvariant ring.
Simplicial dollar game with Dave Perkinson (2022)
The dollar game for graphs is a game played on a graph in which each vertex initially has some number of dollars assigned to it. A move consists of allowing a vertex to either give a dollar to each neighbor or take a dollar from each neighbor. The game is won by arriving at a configuration in which each vertex has a nonnegative amount of dollars. The Riemann-Roch theorem for graphs gives a criterion for when a game is winnable based on the degree, the total amount of money, of a configuration. Its form closely resembles the Riemann-Roch theorem of algebraic geometry. We study a generalization of the dollar game to simplicial complexes introduced by Duval, Klivans, and Martin, and develop an analog of degree for this generalization.