My research interests include all aspects of algebraic combinatorics. One thing I particularly like is the theory of matroids and vector configurations. I like studying them via various algebraic invariants. Here is my CV.

Click on a title below for the corresponding abstract.

Cyclic sieving of finite Grassmannians and flag varieties (with Jia Huang, Discrete Mathematics, Vol 312 (5), 2012).

[ps, pdf, doi ] We prove instances of the cyclic sieving phenonenon for finite partial flag varieties, which carry the action of various tori in GLn(Fq). The formulas for the polynomials involved are sums over minimal length coset representative of certain quotients W/WJ, with a polynomial weight associated to each represenatitive which is a product over its inversions.

Two results on the rank partition of a matroid (Portugal. Math. (N.S.), Vol. 68, Fasc. 4, 2011).

[ps, pdf, doi] We prove two results about the rank partition of a general matroid M. The first says that if there is a (possibly non-standard) tableaux whose rows index independent sets of M then there is a standard tableaux of the same shape with this property. The second result says, roughly, that the rank partition behaves valuatively on matroid polytope subdivisions. Both of these results have a common representation-theoritical motivation, which we present.

Equality of symmetrized tensors and the coordinate ring of the flag variety. (To appear in Linear algebra and its applications.)

[ps, pdf, doi] In this note we give a transparent proof of a result of da Cruz and Dias da Silva on the equality of symmetrized decomposable tensors. This will be done by explaining that their result follows from the fact that the coordinate ring of a flag variety is a unique factorization domain.

Constructions for cyclic sieving phenomena (with S.-P. Eu and V. Reiner. SIAM J. Discrete Math. 25, pp. 1297-1314).

[ps, pdf, doi] We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs.

Tableaux in the Whitney module of a matroid (Séminaire Lotharingien de Combinatoire 63 (2010), Article B63f)

[ps, pdf] 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.

The critical group of a line graph (with A. Manion, M. Maxwell, A. Potechin and V. Reiner. To appear in Annals of Combinatorics)

[ps, pdf] 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.

Products of linear forms and Tutte polynomials (European Journal of Combinatorics Volume 31, Issue 7, (2010), pp. 1924-1935).

[ps, pdf, doi] 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.

A short proof of Gamas's Theorem (Linear Algebra and Its Applications 430 (2009) pp. 791-793).

[ps, pdf, doi] 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 my Ph.D. thesis, entitled Symmetries of tensors. It is a combination of some of the things above, as well as many other things that are yet to be incorporated into published form.

Here is a list of places I have recentlly given or will soon give talks.

March 2012, AMS Special Session on Algebraic Combinatorics. Honolulu.

February 2012, University of Northern Colorado colloquium talk.

January 2012, Syracuse University colloquium talk.

October 2011, AMS Special Session on Algebraic and geometric aspects of matroids. Wake Forest.

March 2011, AMS Special Session on Recent developments in Schubert calculus. Iowa City.

March 2011, AMS Special Session on Algebraic and Geometric Combinatorics.

October 2010, San Francisco State University, AGC Seminar.

June 2010, CMS special session on Algebraic Combinatorics.

April 2010, BAD Math Day.

April 2010, AMS Special Session on Geometric Combinatorics.

March 2010, Unversity of California, Berkeley Combinatorics Seminar.

March 2010, University of Minnesota, 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.