Factoring the characteristic polynomial of a lattice

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.

Applications of quotient posets

“…It was shown in [8] that homogeneous quotients of finite posets are posets. Moreover, it was also shown how the Möbius function behaved when taking quotients.…”

“…When the poset is a lattice there is a canonical choice for the equivalence relation called the standard equivalence relation which was introduced in [8].…”

“…It was shown in [8] that if a partition is induced by a multichain, then assumption (2) of Theorem 4.4 holds if and only if the multichain satisfies the meet condition. We will call a multichain,…”

“…In [8] a class of ranked lattices with this factorization was considered. We wish to give a generalization of these results to arbitrary finite posets with a minimum element.…”

“…We wish to give a generalization of these results to arbitrary finite posets with a minimum element. Many of the theorems from [8] still hold true at this level of generality, but we will need to develop some more concepts in order to show this.…”