Matroid Complex Research Articles

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree 2 n -1-n of the polynomial algebra 𝔽 2 [x 1 ,...,x n ], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k∈ℕ * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of R n,k , we also show that the space of indecomposable elements of 𝔽 q [x 1 ,...,x n ] in degree q n-1 -n has dimension equal to that of a complex cuspidal representation of GL n (𝔽 q ), that is (q-1)(q 2 -1)⋯(q n-1 -1).

Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes


 
 
 A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.
 
 

Finiteness theorems for matroid complexes with prescribed topology

There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=k for any natural number k. In this paper we study the question at the other end of the h-vector: Are there only finitely many (d−1)-dimensional simplicial complexes with hd=k for any given k? The answer is no if we consider general complexes, but we focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. Surprisingly, the answer is yes in all three cases.

Level property of ordinary and symbolic powers of Stanley-Reisner ideals

In this paper, we prove that the t-th ordinary and/or symbolic power of a Stanley-Reisner ideal is level for some positive integer t≥3 if and only if IΔ is a complete intersection and equi-generated. For t=2, we give a characterization of level property of the second symbolic power IΔ(2) when Δ is a matroid complex of dimension one.

Tverberg-Type Theorems for Matroids: A Counterexample and a Proof

Barany, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex into ℝd, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result — but we provide a counterexample already for the case of k = 2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach.

Regularity of symbolic powers and arboricity of matroids

Abstract Let Δ be a matroid complex. In this paper, we explicitly compute the regularity of all the symbolic powers of its Stanley–Reisner ideal in terms of combinatorial data of Δ. In order to do that, we provide a sharp bound between the arboricity of Δ and the circumference of its dual Δ * {\Delta^{*}} .

Möbius coinvariants and bipartite edge-rooted forests

The Möbius coinvariant μ⊥(G) of a graph G is defined to be the Möbius invariant of the dual of the cycle matroid of G. This invariant is known to equal the rank of the reduced homology of the cycle matroid complex of G. For a complete graph Km+1, W. Kook gave an interpretation of μ⊥(Km+1) as the number of edge-rooted forests in Km. In this paper, we obtain a new combinatorial interpretation of μ⊥(Km+1,n+1) as the number of B-edge-rooted forests in Km,n, which is a bipartite analogue of the previous result.Based on these interpretations, we will give new bijective proofs of the formulas for μ⊥(Km+1) and μ⊥(Km+1,n+1) given by I. Novik, A. Postnikov, and B. Sturmfels in terms of the Hermite polynomials. In addition, we will construct a homology basis for the cycle matroid complex of Km+1,n+1 indexed by the B-edge-rooted forests. Also we will discuss the Möbius coinvariant of bi-coned graphs which generalize complete bipartite graphs.

A formula for simplicial tree-numbers of matroid complexes

We give a formula for the simplicial tree-numbers of the independent set complex of a finite matroid M as a product of eigenvalues of the total combinatorial Laplacians on this complex. Two matroid invariants emerge naturally in describing the multiplicities of these eigenvalues in the formula: one is the unsigned reduced Euler characteristic of the independent set complex and the other is the β-invariant of a matroid. We will demonstrate various applications of this formula including a “matroid theoretic” derivation of Kalai’s simplicial tree-numbers of a standard simplex.

Weighted Tree-Numbers of Matroid Complexes

We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes. Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le $\beta$;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes.

H-Vectors of matroids and logarithmic concavity

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:1.The matroid complex of independent subsets of E.2.The broken circuit complex of M relative to an ordering of E. The first implies a conjecture of Colbourn on the reliability polynomial of a graph, and the second implies a conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based on the geometric formula for the characteristic polynomial of Denham, Garrousian, and Schulze.

Generalized Permutohedra, h-Vectors of Cotransversal Matroids and Pure O-Sequences

Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside an r-dimensional convex polytope and bases of a rank r transversal matroid.

The f-vector of a representable-matroid complex is log-concave

We show that the f-vector of the matroid complex of a representable matroid is log-concave. This proves the representable case of a conjecture made by Mason in 1972.

On the Hilbert Coefficients and Betti Numbers of the Stanley-Reisner Ring of a Matroid Complex

Let S=K[x1,…,xn] be a polynomial ring. Herzog and Zheng conjectured that the i-th Hilbert coefficient of a finitely generated graded Cohen-Macaulay S-module N generated in degree 0 is bounded by the functions of the minimal and maximal shifts in the minimal graded free resolution of N over S and the 0-th Betti number of N. Also, Römer asked whether under the Cohen-Macaulay assumption the i-th Betti number of S/I, where I ⊂ S is a graded ideal, can be bounded by the functions of the minimal and maximal shifts of S/I. In this paper, we provide elementary proofs to establish Herzog and Zheng's conjecture and the upper bound part of Römer's question for the Stanley-Reisner ring of a matroid complex.

Comparison of Multiplicity and Final Betti Number of a Standard Graded K-Algebra

Let S be a polynomial ring over a field K and let R be the Stanley-Reisner ring of a matroid complex. In this paper, as a comparison of multiplicity and final Betti number of R over S, the inequality [Formula: see text] is obtained.

Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways.

On the Broken-Circuit Complex of Graphs

It is well known that the Stanley–Reisner ring of a matroid complex is level; this is an algebraic property between the Cohen–Macaulay and Gorenstein properties. A similar result for broken-circuit complexes is no longer true, even for graphs. We show that the Stanley–Reisner ring of the broken-circuit complex, of the cone of any simple graph, is level.

On the h-Vector of a Lattice Path Matroid

Stanley has conjectured that the h-vector of a matroid complex is a pure M-vector. We prove a strengthening of this conjecture for lattice path matroids by constructing a corresponding family of discrete polymatroids.

Face Ring Multiplicity via CM-Connectivity Sequences

Abstract.The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen–Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all (d−1)-dimensional d-Cohen– Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring via the Cohen–Macaulay connectivity of the skeletons of .

A Relative Laplacian Spectral Recursion

The Laplacian spectral recursion, satisfied by matroid complexes and shifted complexes, expresses the eigenvalues of the combinatorial Laplacian of a simplicial complex in terms of its deletion and contraction with respect to vertex $e$, and the relative simplicial pair of the deletion modulo the contraction. We generalize this recursion to relative simplicial pairs, which we interpret as convex subsets of the Boolean algebra. The deletion modulo contraction term is replaced by the result of removing from the convex set $\Phi$ all pairs of faces in $\Phi$ that differ only by vertex $e$. We show that shifted pairs and some matroid pairs satisfy this recursion. We also show that the class of convex sets satisfying this recursion is closed under a wide variety of operations, including duality and taking skeleta.

G-Elements, finite buildings and higher Cohen–Macaulay connectivity

Chari proved that if Δ is a ( d − 1 ) -dimensional simplicial complex with a convex ear decomposition, then h 0 ⩽ ⋯ ⩽ h ⌊ d / 2 ⌋ [M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997) 3925–3943]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M-vector [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548]. This is proved to be true by showing that the set of pairs ( ω , Θ ) , where Θ is a l.s.o.p. for k [ Δ ] , the face ring of Δ, and ω is a g-element for k [ Δ ] / Θ , is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag h-vector of such spaces similar in spirit to those examined in [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548] for order complexes of geometric lattices. This also leads to connections between higher Cohen–Macaulay connectivity and conditions which insure that h 0 < ⋯ < h i for a predetermined i.