A Generalization of Semimodular Supersolvable Lattices

Stanley [18] introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular lattices with atom decision trees strictly contains the class of supersolvable ones, but their characteristic polynomials still factor for combinatorial reasons. We then apply this notion to prove the factorization of polynomials associated with various hyperplane arrangements having non-supersolvable lattices. 1 Atom decision trees In this section we will introduce our main object of study: atom decision trees. We will show that the characteristic polynomial for a semimodular lattice admitting an atom decision tree has non-negative integral roots. In fact, these roots count the sizes of certain sets of atoms of the lattice. We will also note how the semimodular supersolvable lattices of Stanley [18] have atom decision trees and so are a special case. First, however, we must give some definitions and notation. Any terms not defined can be found described in Stanley’s book [19]. Let L be a lattice with meet and join denoted by ∧ and ∨, respectively. All our lattices will be finite having a minimal element 0̂ and maximal element 1̂. The Möbius function of L is defined inductively on elements x ∈ L by μ(x) = { 1 if x = 0̂ −∑y 0̂. The Möbius function is one of the fundamental invariants of L. Now suppose L is graded with the rank of x ∈ L denoted by rk(x). Then the characteristic polynomial of L is χ(L, t) = ∑ x∈L μ(x)trk(1̂)−rk(x). (1) One uses the corank of x, rather than its rank, as the exponent on t so that the polynomial will be monic. Since the characteristic polynomial is just the generating function for the Möbius function, it is also of fundamental importance. Now suppose L is semimodular so that rk(x) + rk(y) ≥ rk(x ∧ y) + rk(x ∨ y) for all x, y ∈ L (2) and consider the set A of atoms of L. Given a subset B ⊆ A, we define