I study algebraic and geometric combinatorics. I particularly enjoy seeing the combinatorics of matroids reflected in the algebraic and geometric invariants of hyperplane arrangements. I often try to use tools from representation theory, commutative algebra or algebraic geometry to illuminate these invariants.
13. [ps, pdf] Equivariant K-classes of matrix orbit closures (with Alex Fink). Submitted.
We show that the equivariant K-theory class of an orbit closure GLr v Tn of r-by-n matrices depends only on the matroid of the matrix v. As consequences, the same is true of the equivariant cohomology class of this orbit, as well as of the isomorphism types of the cyclic GLr or Sn-module generated by the tensor product of the columns of v. We give formulae for certain coefficients of these matroid invariants, and for the entire invariants in some cases, as well as formulae for the behaviour of these invariants under certain matroid operations. We also give generators for the ideal of the orbit closure up to radical.
12. Ideals generated by superstandard tableaux (with Winfried Bruns and Aldo Conca). To appear in a special volume dedicated to the 2012--2013 MSRI program on Commutative Algebra.
[ps, pdf] We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These bitableaux form a Gr\"obner basis of J, and J has a linear minimal free resolution. These results are used to derive a new generating set for the Grothendieck group of finitely generated (Tm x GLn(K))-equivariant modules over K[X]. We employ the Knuth--Robinson--Schensted correspondence and a toric deformation of the multi-Rees algebra that parameterizes the ideals J.
11. Vanishing of doubly symmetrized tensors (with J.A. Dias da Silva and Amélia Fonseca). Elect. J. Combinatorics, Vol 20 (2), P60, 9pp, 2013.
[ps, pdf] Symmetrizations of tensors by irreducible characters of the symmetric group serve as natural analogues of symmetric and skew-symmetric tensors. The question of when a symmetrized decomposable tensor is non-zero is intimately related to the rank partition of a matroid extracted from the tensor. In this paper we characterize the non-vanishing of the symmetrization of certain partially symmetrized decomposable tensors. Our answers are phrased in terms of rank partitions of matroids.
10. Extending the parking representation (with Brendon Rhoades). Journal of Combinatorial Theory, Series A, 123 (1), (2014), 43-56.
[ps, pdf] We prove that the action of Sn on the vector space spanned by parking functions of length n extends to an action of Sn+1. More precisely, we construct a graded Sn+1-module whose restriction to Sn is isomorphic to the parking representation. We describe the Sn-Frobenius characters on our module in all degrees and the Sn+1-Frobenius character in extreme degrees. We give a bivariate generalization of our module whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.
Here is some Mathematica code that checks whether a given representation of Sm is the restriction of a representation of Sn.
9. Critical groups of graphs with reflective symmetry. J. Algebraic Combinatorics, 2013.
[ps, pdf, doi] The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. For a graph G with a certain reflective symmetry, we generalize a result of Ciucu-Yan-Zhang factorizing the spanning tree number of G by interpreting this as a result about the critical group of G. Our result takes the form of an exact sequence, and explicit connections to bicycle spaces are made.
- 8. 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.
- 7. 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.
6. Equality of symmetrized tensors and the coordinate ring of the flag
variety. Linear algebra and its applications, 438(2):658-662, 2013.
[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.
- 5. 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.
4. Tableaux in the Whitney module of a matroid (Seminaire 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.
3. The critical group of a line graph (with A. Manion, M. Maxwell,
A. Potechin and V. Reiner. To appear in Annals of
[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.
2. Products of linear forms and Tutte polynomials (European Journal of
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.
- 1. 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.