The Dowling Transform of Subspace Arrangements

We define the Dowling transform of a real frame arrangement and show how the characteristic polynomial changes under this transformation. As a special case, the Dowling transform sends the braid arrangement An to the Dowling arrangement. Using Zaslavskyʹs characterization of supersolvability of signed graphs, we show supersolvability of an arrangement is preserved under the Dowling transform. We also give a direct proof of Zaslavskyʹs result on the number of chambers and bounded chambers in a real hyperplane arrangement.