isDecomposable(CentralArrangement,Ring) -- whether a hyperplane arrangement decomposable in the sense of Papadima-Suciu

Synopsis

Function: isDecomposable
Usage:

isDecomposable(A, R)

isDecomposable A
Inputs:
- A, a central hyperplane arrangement,
- R, a ring, an optional coefficient ring used as the coefficient field for the holonomy Lie algebra. If unspecified, R=QQ
Outputs:
- a Boolean value, that is true if the hyperplane arrangement decomposes in the sense of Papadima and Suciu over the given coefficient field

Description

Following Definition 2.3 in Stefan Papadima and Alexander I. Suciu's paper "When does the associated graded Lie algebra of an arrangement group decompose?", Commentarii Mathematici Helvetici (2006) 859-875, arXiv:math/0309324, a hyperplane arrangement is decomposable if the derived subalgebra of its holonomy Lie algebra is a direct sum of the derived subalgebras of free Lie algebras, indexed by the rank-2 flats of the arrangement.

As described in the introduction of Papadima-Suciu, the X3 arrangement is decomposable. The hyperplane arrangement defined by a type $A_3$ root system is not decomposable. The authors show that a graphic arrangement is decomposable over ${\mathbb Q}$ if and only if it is decomposable over any other field. In general, it is not known if there exist arrangements for which property of being decomposable depends on the choice of field.

i1 : X3 = arrangement "X3"

o1 = {x , x , x , x  + x , x  + x , x  + x }
       1   2   3   1    2   1    3   2    3

o1 : Hyperplane Arrangement

i2 : assert isDecomposable X3

i3 : assert isDecomposable(X3, ZZ/5)

i4 : assert not isDecomposable typeA 3

Ways to use this method: