Fix an irreducible crystallographic root system and its simple system. Then we can define
heights of the positive roots. It is classically known that, if we consider the dual partition of
the height disrtibution of this positive roots, then they coincide with the exponents of the
original root system. To this interesting facts (of course easy to show if we use the classification),
several mathematicians like Kostant or Macdonald gave proofs without using the classification
of the root systems.

In this talk, we give a different proof to this dual partition theorem by using the freeness of
hyperplane arrangements. So the proof is algebraic, algebro-geometric and combinatorial. Also,
we can generalize the original dual partition theorem to the category of ideals in the root
poset.

This is the joint work with Mohamed Barakat, Michael Cuntz, Torsten Hoge and Hiroaki Terao.