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

TL;DR

This paper studies solvable and nilpotent matroids, focusing on their realizability, associated varieties, and algebraic properties, providing conditions for irreducibility and explicit generators for their ideals.

Contribution

It introduces new classes of matroids, analyzes their realization spaces, and offers explicit algebraic descriptions of their associated varieties and ideals.

Findings

Nilpotent matroid varieties are realizable and irreducible.

Certain solvable paving matroid varieties are proven irreducible.

Complete generating sets for matroid ideals of forest configurations.

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.