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Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties

Emiliano Liwski, Fatemeh Mohammadi

TL;DR

The work introduces and analyzes two new matroid families, nilpotent and solvable, to study realizability and the irreducible decomposition of their realization spaces $\Gamma_{M}$ and associated varieties $V_{M}$ and $V_{\mathcal{C}(M)}$. It develops liftability techniques and Grassmann-Cayley algebra to derive defining equations and constructs a finite generating set $G_{M}$ for forest configurations, providing $I_{M}$ up to radical in key cases. The authors prove realizability and irreducibility results for nilpotent matroids and establish irreducibility for broad classes of solvable paving matroids, with explicit results for nilpotent paving matroids without deg$>2$ points and forest configurations. Applications to hypergraph varieties and their decompositions illustrate connections to determinantal ideals and conditional independence models, offering new tools for understanding the geometry of matroid-related spaces. Overall, the paper deepens the link between combinatorial matroid structure and algebraic geometry, furnishing concrete generating sets and irreducibility criteria that inform both theory and computation.

Abstract

We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their associated matroid and circuit varieties. Additionally, we describe a finite generating set for the corresponding ideals, considered up to radical. We establish sufficient conditions for both the realizability of these matroids and the irreducibility of their associated varieties. Specifically, we establish the realizability and irreducibility of matroid varieties associated with nilpotent matroids and prove the irreducibility of matroid varieties arising from certain classes of solvable paving matroids. Additionally, we analyze the defining polynomial equations of these varieties using Grassmann-Cayley algebra and geometric liftability techniques. Furthermore, we provide a complete generating set for the matroid ideals associated with forest configurations.

Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties

TL;DR

The work introduces and analyzes two new matroid families, nilpotent and solvable, to study realizability and the irreducible decomposition of their realization spaces and associated varieties and . It develops liftability techniques and Grassmann-Cayley algebra to derive defining equations and constructs a finite generating set for forest configurations, providing up to radical in key cases. The authors prove realizability and irreducibility results for nilpotent matroids and establish irreducibility for broad classes of solvable paving matroids, with explicit results for nilpotent paving matroids without deg points and forest configurations. Applications to hypergraph varieties and their decompositions illustrate connections to determinantal ideals and conditional independence models, offering new tools for understanding the geometry of matroid-related spaces. Overall, the paper deepens the link between combinatorial matroid structure and algebraic geometry, furnishing concrete generating sets and irreducibility criteria that inform both theory and computation.

Abstract

We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their associated matroid and circuit varieties. Additionally, we describe a finite generating set for the corresponding ideals, considered up to radical. We establish sufficient conditions for both the realizability of these matroids and the irreducibility of their associated varieties. Specifically, we establish the realizability and irreducibility of matroid varieties associated with nilpotent matroids and prove the irreducibility of matroid varieties arising from certain classes of solvable paving matroids. Additionally, we analyze the defining polynomial equations of these varieties using Grassmann-Cayley algebra and geometric liftability techniques. Furthermore, we provide a complete generating set for the matroid ideals associated with forest configurations.
Paper Structure (18 sections, 29 theorems, 43 equations, 3 figures, 1 algorithm)

This paper contains 18 sections, 29 theorems, 43 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Realization spaces of matroids and matroid varieties
  5. Grassmann-Cayley algebra
  6. Nilpotent matroids
  7. Definition and basic properties
  8. Realizability and irreducibility of nilpotent realization spaces
  9. Defining equations
  10. Nilpotent paving matroids without points of degree greater than two
  11. Forest configurations
  12. Defining equations
  13. Solvable matroids
  14. Definition and basic properties
  15. Paving matroids without points of degree greater than two
  16. ...and 3 more sections

Key Result

Theorem (A)

Let $M$ be a nilpotent matroid and let $\Gamma_{M},V_{M}$ and $V_{\mathcal{C}(M)}$ denote its realization space, matroid variety and circuit variety, respectively, as defined in Subsection sec reali. Then the following statements hold:

Figures (3)

  • Figure 1: (Left) Three concurrent lines; (Center) Quadrilateral set; (Right) Pascal configuration.
  • Figure 2: (Left) a nilpotent matroid; (Center) a forest configuration; (Right) Fano plane (not solvable).
  • Figure 3: Minimal matroids of Fano configuration.

Theorems & Definitions (92)

  • Definition 1.1: Definitions \ref{['k2']} and \ref{['m7']}
  • Theorem (A)
  • Theorem (B)
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • ...and 82 more