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

We introduce the families of solvable and nilpotent matroids, and study their realization spaces and their closures. Specifically, we analyze their associated varieties and their irreducible decompositions. Additionally, we study a subfamily of nilpotent matroids, called weak nilpotent matroids, and compute a finite set of defining equations of their associated matroid varieties.