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Orbits and self-twuality in set systems and delta-matroids

Zhuo Li, Xian'an Jin, Qi Yan

TL;DR

The paper develops a unified algebraic framework for self-twuality in set systems and delta-matroids by constructing a group action of the semidirect product $\mathcal{G}^n \rtimes_\phi S_n$ (with $\mathcal{G} \cong S_3$) on ground sets. It proves that self-twuality properties propagate along orbits and provides a multimatroid-based orbit characterization for vf-safe delta-matroids, linking their structure to tight 3-matroids. It further shows that the orbit of ribbon-graphic delta-matroids can be described in terms of the tight 3-matroid associated with the medial graph of the corresponding ribbon graph, thereby unifying and extending prior results in embedded-graph theory. The work answers an open question of Abrams and Ellis-Monaghan and offers a robust algebraic toolkit for analyzing self-twuality across delta-matroids, ribbon graphs, and their generalizations.

Abstract

We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group framework of Abrams and Ellis-Monaghan from ribbon graphs to set systems, facilitating a systematic investigation of self-twuality. We prove that different forms of self-twuality propagate through orbits under the group action and establish a characterization of the orbit of a vf-safe delta-matroid via multimatroids. As an application, we analyze orbits of ribbon-graphic delta-matroids. Our work answers a question posed by Abrams and Ellis-Monaghan and provides a unified algebraic framework for studying self-twuality in combinatorial structures.

Orbits and self-twuality in set systems and delta-matroids

TL;DR

The paper develops a unified algebraic framework for self-twuality in set systems and delta-matroids by constructing a group action of the semidirect product (with ) on ground sets. It proves that self-twuality properties propagate along orbits and provides a multimatroid-based orbit characterization for vf-safe delta-matroids, linking their structure to tight 3-matroids. It further shows that the orbit of ribbon-graphic delta-matroids can be described in terms of the tight 3-matroid associated with the medial graph of the corresponding ribbon graph, thereby unifying and extending prior results in embedded-graph theory. The work answers an open question of Abrams and Ellis-Monaghan and offers a robust algebraic toolkit for analyzing self-twuality across delta-matroids, ribbon graphs, and their generalizations.

Abstract

We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group framework of Abrams and Ellis-Monaghan from ribbon graphs to set systems, facilitating a systematic investigation of self-twuality. We prove that different forms of self-twuality propagate through orbits under the group action and establish a characterization of the orbit of a vf-safe delta-matroid via multimatroids. As an application, we analyze orbits of ribbon-graphic delta-matroids. Our work answers a question posed by Abrams and Ellis-Monaghan and provides a unified algebraic framework for studying self-twuality in combinatorial structures.
Paper Structure (6 sections, 13 theorems, 102 equations, 1 figure)

This paper contains 6 sections, 13 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Delta-matroids and fundamental operations
  3. Group actions on set systems
  4. Propagation of self-twual in orbits
  5. Orbits characterization via multimatroids
  6. Orbits of ribbon-graphic delta-matroids

Key Result

Proposition 3.1

Let $\phi: S_n \rightarrow \operatorname{Aut}(\mathcal{G}^n)$ be defined by $\phi(\pi) \mapsto \phi_\pi$, where $\phi_\pi(\hat{g}) = \hat{g}\pi^{-1}$. Then $\phi$ is a homomorphism, and the semidirect product $\mathcal{G}^n \rtimes_\phi S_n$ acts on $\mathcal{D}_{(n)}$ by $(\hat{g},\pi)D = \hat{g} \

Figures (1)

  • Figure 1: A vertex of $G_m$ and its three vertex transitions

Theorems & Definitions (34)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3: Chun2019
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.1
  • proof
  • proof
  • ...and 24 more