Arrangements of ideal type

In the same paper from 2006, Sommers and Tymoczko define a Poincaré polynomial I(t) associated with each ideal I which generalizes the Poincaré polynomial W(t) for the underlying Weyl group W. Solomon showed that W(t) satisfies a product decomposition depending on the exponents of W for any Coxeter group W. Sommers and Tymoczko showed in a case by case analysis in types An, Bn and Cn, and some small rank exceptional types that a similar factorization property holds for the Poincaré polynomials I(t) generalizing the formula of Solomon for W(t). They conjectured that their multiplicative formula for I(t) holds in all types. In our second aim to investigate this conjecture further, the same inductive tools we develop to obtain inductive freeness of the AI are also employed. Here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions.