My research interests include all aspects of algebraic combinatorics, in particular the theory of matroids and combinatorial representation theory. Currently, I am studying matroids and vector configurations by associating group representations to them.
[ps, pdf] Tableaux in the Whitney Module of a Matroid (Draft) The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of the matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom (also known as Schubert, or shifted) matroids. We will also describe a basis for the doubly multilinear submodule of the Whitney module spanned by hook shaped tableaux.
[ps, pdf] The Critical Group of a Line Graph (with A. Manion, M. Maxwell, A. Potechin and V. Reiner. Submitted) The critical group of a graph is the torsion subgroup of the cokernel of its Laplacian matrix. Its order is the number of spanning forests in the graph. This paper investigates how the critical group of a graph is related to that of its line graph. The main results bound the number of generators and give strong restraints on the p-primary structure of the critical group of a line graph. When the graph is regular or semiregular we give additional structural results.
[ps, pdf] Products of Linear Forms and Tutte Polynomials (Submitted). This paper studies the vector space spanned by products of linear forms from a fixed set Δ. Using a result of Orlik and Terao we obtain a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(Δ;1+x,y). By specializing x and y we obtain various results from the literature.
[ps, pdf] A short proof of Gamas's Theorem (Linear Algebra and Its Applications 430 (2009) pp. 791-793). This is a short and self-contained proof of Gamas's Theorem on the vanishing of symmetrized tensors. For my purposes, this result states under what conditions an irreducible representation of GL(V) appears in the smallest GL(V) representation containing a fixed decomposable tensor.
Here is a list of places I have given or will give talks.
June 2010, CMS special session on Algebraic Combinatorics.
October 2009, University of California, Davis Discrete Math Seminar.
April 2009, AMS special session on Matroids in Algebra and Geometry.
January 2009, University of Illinois Algebra-Geometry-Combinatorics Seminar.
October 2009, l'Université Pierre et Marie Curie.
October 2009, Centro de Estruturas Lineares e Combinatórias.
October 2009, AMS special session on Combinatorial Representation Theory.
September 2009, University of Kansas Combinatorics Seminar.
[ps, pdf] Critical groups of some regular line graphs. This paper arose out of the Summer 2003 REU program at the University of Minnesota. The critical group of the line graph of the complete bipartite graph is given. Conjectures are also presented that form the basis for future work were the critical group of a graph is related to that of its line graph. This paper is kept here since there is nowhere else for it to go.