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Matroids from gain graphs over quotient groups

Zach Walsh

Abstract

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $Γ$ with Frobenius kernel $Γ_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $Γ/Γ_1$. While the hypothesis that $Γ$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $Γ$ is any finite group with a nontrivial proper normal subgroup $Γ_1$ and there is a construction that takes in a complete $Γ$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(Γ/Γ_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $Γ$-balanced, then $Γ$ is a Frobenius group with Frobenius kernel $Γ_1$.

Matroids from gain graphs over quotient groups

Abstract

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group with Frobenius kernel and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group . While the hypothesis that is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if is any finite group with a nontrivial proper normal subgroup and there is a construction that takes in a complete -gain graph and outputs an elementary lift of the frame matroid of the underlying -gain graph so that a cycle of the graph is a circuit of if and only if it is -balanced, then is a Frobenius group with Frobenius kernel .
Paper Structure (23 sections, 28 theorems, 14 equations, 5 figures)

This paper contains 23 sections, 28 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theory
  4. Semidirect products
  5. Frobenius groups
  6. Group partitions
  7. Infinite groups
  8. Gain graphs
  9. Biased graphs and their matroids
  10. The construction
  11. Properties
  12. Rank, bases, and circuits
  13. Minors
  14. Duality and direct sums
  15. Examples
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a group, let $\Gamma_1$ be a normal subgroup of $\Gamma$, and let $\{\Gamma_1\} \cup \mathcal{A}$ be a partition of $\Gamma$ so that if $A \in \mathcal{A}$ then $A$ is malnormal and every conjugate of $A$ is in $\mathcal{A}$. Then every $\Gamma$-gain graph $(G, \psi)$ has a canonical

Figures (5)

  • Figure 1: Image (i) is a $\Gamma$-gain graph, and (ii) is the $\Gamma$-gain graph obtained by switching at $v$ with value $\gamma$.
  • Figure 2: Image (a) shows a $\mathbb Z$-gain graph $(G, \psi)$ with one balanced cycle, and (b) shows the $(\mathbb Z/2\mathbb Z)$-gain graph $(G, \psi/2\mathbb Z)$, which has three balanced cycles.
  • Figure 3: Theta graphs, tight handcuffs, and loose handcuffs are subdivisions of the graphs (a), (b), and (c), respectively.
  • Figure 4: If $\mathbb F$ is a field and $\Gamma_1$ and $\Gamma_2$ are subgroups of $\mathbb F^+$ and $\mathbb F^{\times}$, respectively, so that that $\Gamma_1$ is closed under scaling by elements in $\Gamma_2$, then the class of matroids with an $\mathbb F$-representation of the form shown above is minor-closed.
  • Figure 5: A $(\mathop{\mathrm{GF}}\nolimits(5)^{+} \rtimes \mathop{\mathrm{GF}}\nolimits(5)^{\times})$-gain graph $(G, \psi)$ with an orientation $D$ of $G$, and the incidence matrix $A(D, \psi)$ over $\mathop{\mathrm{GF}}\nolimits(5)$.

Theorems & Definitions (97)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • ...and 87 more