Factoring the characteristic polynomial of a lattice

TL;DR

This paper presents a new method based on poset quotients to determine when the characteristic polynomial of certain finite lattices factors into nonnegative integer roots, linking it to graph invariants and extending existing theorems.

Contribution

Introduces a novel approach using poset quotients to establish conditions for polynomial factorization and connects it to graph theory applications.

Findings

Provides two simple conditions for polynomial factorization with nonnegative integer roots.

Shows Stanley's Supersolvability Theorem is a corollary of the new method.

Establishes equivalence of three conditions for factorization.

Abstract

We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers. This method is based on the notion of a quotient of a poset which will be developed to explain this factorization. Our main theorem will give two simple conditions under which the characteristic polynomial factors with nonnegative integer roots. We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization. To our knowledge, all other theorems in this area only give conditions which imply factorization. This theorem will be used to connect the generating function for increasing spanning forests of a graph to its chromatic polynomial. We finish by mentioning some other applications of quotients of posets as well as some open questions.