Matroids with different configurations and the same G-invariant

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Abstract

From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the G-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same G-invariant. We offer several such constructions along with tools for developing more.

Introduction

The configuration of a matroid, which Eberhardt [9] introduced, is obtained from its lattice of cyclic flats (that is, flats that are unions of circuits) by recording the abstract lattice structure along with just the size and rank of each cyclic flat, not the set. Eberhardt proved that from the configuration of a matroid M, one can compute its Tutte polynomial,T(M;x,y)=AE(M)(x1)r(M)r(A)(y1)|A|r(A). The data recorded in the Tutte polynomial is the multiset of size-rank pairs, (|A|,r(A)), over all AE(M). The Tutte polynomial is one of the most extensively studied invariants of a matroid (see, e.g., [6], [10]).

Strengthening Eberhardt's result, Bonin and Kung [2, Theorem 7.3] showed that from the configuration of a matroid M, one can compute its G-invariant, G(M). Derksen [7] introduced the G-invariant and showed that the Tutte polynomial can be computed from it. The perspective on the G-invariant that we use is the reformulation introduced in [2]: G(M) records the multiset of sequences of sizes of the sets in flags (maximal chains of flats) of M. (Section 2 gives a more precise formulation.) This information about flags just begins to suggest the wealth of information that G(M) captures beyond what the Tutte polynomial contains; other examples (from [2, Section 5]) include the number of saturated chains of flats with specified sizes and ranks, the number of circuits and cocircuits of each size (in particular, the number of spanning circuits), and, for each triple (m,k,c) of integers, the number of flats F with |F|=m and r(F)=k for which the restriction M|F has c coloops. Beyond the multiset of size-rank pairs (|A|,r(A)) noted above as equivalent to the Tutte polynomial, as [3, Theorem 5.3] shows, from G(M), one can find, for each triple (m,k,c) of integers, the number of sets A with |A|=m and r(A)=k for which the restriction M|A has c coloops. Akin to the universality property of the Tutte polynomial among matroid invariants that satisfy deletion-contraction rules, Derksen and Fink [8] showed that the G-invariant is a universal valuative invariant for subdivisions of matroid base polytopes.

Reflecting on the proof of [2, Theorem 7.3] reveals that the chains of cyclic flats in the configuration play the key role; the lattice structure is secondary. So if one can construct pairs of non-isomorphic lattices for which one can relate their chains via bijections, one might be able to produce pairs of matroids with different configurations and the same G-invariant. That is the idea that we develop in this paper and use to shed light on how much stronger the configuration is compared to the G-invariant. Since the Tutte polynomial can be computed from the G-invariant, our results also contribute to the theory of Tutte-equivalent matroids (see [5]).

In Section 2, we review the relevant background and previously known examples of matroids with different configurations but the same G-invariant. The core of the paper is Section 3, where we develop the tools that we apply in Sections 4 and 5 to give the constructions of interest. These tools and the strategy in the proofs of Theorem 4.1, Theorem 5.1 are likely to be useful to obtain more such constructions.

Section snippets

Notation, background, and prior results

Our notation and terminology for matroid theory follow Oxley [11]. Let [n] denote the set {1,2,,n}. Let 0ˆ denote the least element of a lattice L, and let 1ˆ denote its greatest element. If we need to clarify in which lattice an interval is formed, we use a subscript, as in [a,b]L. Likewise, we may use 0ˆL and 1ˆL. For a lattice L, we let L denote L with 0ˆ and 1ˆ removed, so L is the open interval (0ˆ,1ˆ).

Given a matroid M, a subset A of E(M) is cyclic if A is a (possibly empty) union of

The tools

The first lemma generalizes an argument from the proof of [2, Theorem 7.3]. The join F1F2 mentioned below is the join in the lattice Z(M) of cyclic flats.

Lemma 3.1

Assume that the function g:Z(M)22E(M) has the property that if g(F1)g(F2), then F1F2Z(M) and g(F1)g(F2)g(F1F2). Then|FZ(M)g(F)|=SZ(M),S(1)|S|+1|FSg(F)|=nonempty chains SZ(M)(1)|S|+1|FSg(F)|.

Before proving the lemma, we note an example of such a function g. For a matroid M, let gM:Z(M)22E(M) be given bygM(F)={X:XE

Application: a construction modifying a lattice

In this section we show how to extend a finite lattice in different ways to produce lattices in different configurations that yield the same G-invariant.

We first construct the lattices that appear in Theorem 4.1. Let a1,a2,,am be distinct elements of a finite lattice L for which

  • there is a bL with aiaj=b for all i,j[m] with ij, and

  • for each i[m1], there is a lattice isomorphism τm,i:[0ˆ,am][0ˆ,ai] with τm,i(y)=y for all y[0ˆ,b].

Set τi,m=τm,i1 and, for distinct i,j[m1], set τi,j=τm,jτ

Application: a construction using paving matroids

In Theorem 5.1 we show how to use the lattices of flats of paving matroids as the lattices in different configurations that yield the same G-invariant. Examples are given after the proof.

Theorem 5.1

Let N1 and N2 be non-isomorphic rank-r paving matroids on a set E. For i[2], let Hi be the set of hyperplanes of Ni and let Li be the lattice of flats of Ni. Assume that there are partitions {A1,A2,,Ap} of H1H2 and {B1,B2,,Bp} of H2H1, and, for each j[p], a bijection αj:EE such that XAj if and only if αj

Acknowledgements

The author thanks both referees for their careful reading of the manuscript and their thoughtful comments.

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