Arrangements of ideal type

作者：Roehrle Gerhard

来源：Journal of Algebra, 2017, 484: 126-167.

DOI：10.1016/j.jalgebra.2017.04.008

• Root system
• Weyl arrangement
• Arrangement of ideal type
• Free arrangement
• Inductively free arrangement
• Supersolvable arrangement
• Inductively factored arrangement

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A(I) stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A(I) is free if the root system is of classical type or G(2) and conjectured that this is also the case for all types. This was established only very recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A(I) is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro-Steinberg-Kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated Weyl group.
Our first aim in this paper is to investigate a stronger freeness property of the A(I). We show that all A(I) are inductively free, with the possible exception of some cases in type E-8.
In the same paper from 2006, Sommers and Tymoczko define a Poincare polynomial I(t) associated with each ideal I which generalizes the Poincare 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 A(n), B-n, and C-n, and some small rank exceptional types that a similar factorization property holds for the Poincare 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 A(I) are also employed. Here we also show that this conjecture holds in-ductively in almost all instances with only a small number of possible exceptions.

• 出版日期2017-8-15