Super solvability and Freeness for 𝝍 Graphical Arrangements

Abstract

We present a self-contained exposition of super solvability and freeness for 𝜓-graphical arrangements, including full statements of the main characterization theorems, proof sketches, worked examples, and a short discussion of methods used (modular chains, vertex-weighted graphs, and the addition–deletion techniques for free arrangements). The paper collects and arranges the results that answer Stanley's conjectures on 𝜓-graphical arrangements: the characterization of super solvability due to Mu & Stanley and the subsequent resolution of the freeness conjecture by Suyama & Tsujie. We highlight the key combinatorial and algebraic tools (chordality, modular chains in intersection lattices, and Terao's addition–deletion framework), indicate how they are applied in this context, and point out directions for further research.