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A q-analogue of delta-matroids and related concepts

Michela Ceria, Trygve Johnsen, Relinde Jurrius

TL;DR

This work builds $q$-analogues of Δ-matroids and g-matroids by working over finite fields with subspace ground structures. It defines $q$-$ abla$-matroids via codimension-1 exchange axioms and explores duality, strong maps, and the relationship to $q$-matroids, $q$-g-matroids, and restriction issues. The paper proves that $q$-g-matroids are $q$-Δ-matroids, studies saturation and equivalence questions, and connects these structures to rank-metric/code representations through strong maps and rank formulas. It also outlines representability prospects via pairs of codes and introduces a rank theory for $q$-Δ-matroids, highlighting both parallels and substantial obstacles relative to the classical (set-based) theory. Overall, the work lays foundational language and results for q-analogue matroid theory with potential applications to coding theory and algebraic geometry on surfaces, while identifying major open questions on operations like restriction/contraction and strong-map equivalences.

Abstract

We define and study q-delta-matroids, and q-g-matroids. These objects are analogues, for finite-dimensional vector spaces over finite fields, of delta-matroids and g-matroids arising from finite sets. We compare axiomatic descriptions with definitions by means of strong maps of q-matroids.

A q-analogue of delta-matroids and related concepts

TL;DR

This work builds -analogues of Δ-matroids and g-matroids by working over finite fields with subspace ground structures. It defines --matroids via codimension-1 exchange axioms and explores duality, strong maps, and the relationship to -matroids, -g-matroids, and restriction issues. The paper proves that -g-matroids are -Δ-matroids, studies saturation and equivalence questions, and connects these structures to rank-metric/code representations through strong maps and rank formulas. It also outlines representability prospects via pairs of codes and introduces a rank theory for -Δ-matroids, highlighting both parallels and substantial obstacles relative to the classical (set-based) theory. Overall, the work lays foundational language and results for q-analogue matroid theory with potential applications to coding theory and algebraic geometry on surfaces, while identifying major open questions on operations like restriction/contraction and strong-map equivalences.

Abstract

We define and study q-delta-matroids, and q-g-matroids. These objects are analogues, for finite-dimensional vector spaces over finite fields, of delta-matroids and g-matroids arising from finite sets. We compare axiomatic descriptions with definitions by means of strong maps of q-matroids.
Paper Structure (16 sections, 31 theorems, 15 equations, 1 figure)

This paper contains 16 sections, 31 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\Delta$-matroids
  4. $q$-matroids
  5. Strong maps
  6. $q$-$\Delta$-matroids
  7. Defining the $q$-analogue of a $\Delta$-matroid.
  8. Examples from $q$-matroids
  9. $q$-$\Delta$-matroids and restriction: some examples
  10. A $q$-analogue of $g$-matroids and weak $g$-matroids
  11. $q$-$g$-matroids
  12. Relation to deltamatroids
  13. Partial results about equivalences
  14. Representability and rank
  15. Pairs of Codes
  16. ...and 1 more sections

Key Result

Proposition 2.2

The pair $\Delta^*=(E,\mathcal{F}^*)$ is a $\Delta$-matroid, if $\mathcal{F}^*$ is the family of set-theoretical complements of the sets that are members of $\mathcal{F}$, for a $\Delta$-matroid $\Delta=(E,\mathcal{F})$. The $\Delta$-matroid $\Delta^*=(E,\mathcal{F}^*)$ is called the dual $\Delta$-m

Figures (1)

  • Figure 1: Relations between various structures. An arrow reads "is a". The '?' indicates that we do not know if this relation is true.

Theorems & Definitions (84)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.8
  • ...and 74 more