Search

For a matroid, we define a new simplicial complex whose facets are indexed by its independent sets. This complex contains the external activity complex as a subcomplex. We call our complex the augmented external activity complex since its definition is motivated by the recently defined augmented tautological classes of matroids. We prove that our complex is shellable and show that our shelling satisfies the stronger property of being an $H$-shelling. This explicates our result that the $h$-vector of our complex is the $f$-vector of the independence complex. We also define an augmented no broken circuit complex, which contains the usual no broken circuit complex as a subcomplex. We prove its shellability and show that our shelling is also an $H$-shelling. The $h$-vector of this complex is the $f$-vector of the no broken circuit complex., Comment: Minor changes from v1. To appear Journal of Combinatorics

We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory., Comment: 71 pages; comments welcome. v4: minor edits. To appear in Invent. Math

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson., Comment: 37 pages. To appear DMJ

The group $G = GL_r(k) \times (k^\times)^n$ acts on $\mathbf{A}^{r \times n}$, the space of $r$-by-$n$ matrices: $GL_r(k)$ acts by row operations and $(k^\times)^n$ scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in $G$ equivariant $K$-theory of $\mathbf{A}^{r \times n}$ is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities., Comment: 27pp. Expanded introduction. New section with positivity conjectures for all matroids. Comments welcome!

The group $G(m,1,n)$ consists of $n$-by-$n$ monomial matrices whose entries are $m$th roots of unity. It is generated by $n$ complex reflections acting on $\mathbf{C}^n$. The reflecting hyperplanes give rise to a (hyperplane) arrangement $\mathcal{G} \subset \mathbf{C}^n$. The internal zonotopal algebra of an arrangement is a finite dimensional algebra first studied by Holtz and Ron. Its dimension is the number of bases of the associated matroid with zero internal activity. In this paper we study the structure of the internal zonotopal algebra of the Gale dual of the reflection arrangement of $G(m,1,n)$, as a representation of this group. Our main result is a formula for the top degree component as an induced character from the cyclic group generated by a Coxeter element. We also provide results on representation stability, a connection to the Whitehouse representation in type~A, and an analog of decreasing trees in type~B., Comment: 21 pages, 1 figure. Corrected statement of the theorem and proof

Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is in the variety., Comment: 11 pages. Theorem 3.5 here proves a version of the main result of arXiv:1306.1810v5, the proof of which contained an error

Let $G$ be the group $GL_r(C) \times (C^\times)^n$. We conjecture that the finely-graded Hilbert series of a $G$ orbit closure in the space of $r$-by-$n$ matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the $GL_r(C)$ variables are determined by the matroid, and that the orbit closure has a set-theoretic system of ideal generators whose combinatorics are also so determined. We also discuss relations between these Hilbert series for related matrices, including their stabilizing behaviour as $r$ increases., Comment: The proof of the main theorem (Theorem 5.1) of v5, asserting the invariance of the equivariant K-class of a matrix orbit closure, was incorrect and has been removed

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 (T_m x GL_n(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., Comment: 16 pages

The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $Park_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ 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., Comment: 18 pages, 6 figures

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., Comment: 17 pages. To appear in J. Alg. Comb

In this paper we prove instances of the cyclic sieving phenomenon for finite Grassmannians and partial flag varieties, which carry the action of various tori in the finite general linear group GL_n(F_q). The polynomials involved are sums of certain weights of the minimal length parabolic coset representatives of the symmetric group S_n, where the weight of a coset representative can be written as a product over its inversions., Comment: 18 pages

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., Comment: 5 pages

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., Comment: 18 pages, typos fixed, to appear in SIAM J. Discrete Math

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 a 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 matroids (also known as Schubert, or shifted matroids). The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity of hook shapes in this representation in terms of no broken circuit sets.

Let \Delta be a finite sequence of n vectors from a vector space over any field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v \in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(\Delta;1+x,y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries., Comment: Minor changes. Accepted for publication in European Journal of Combinatorics

If \chi^\lambda is the irreducible character of the symmetric group S_n corresponding to the partition \lambda of n then we may symmetrize a tensor v_1 \otimes ... \otimes v_n by \chi^\lambda. Gamas's theorem states that the result is not zero if and only if we can partition the set {v_i} into linearly independent sets whose sizes are the parts of the transpose of \lambda. We give a short and self-contained proof of this fact.

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph. The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices. The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups. The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion., Comment: 36 pages