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Europ. J. Combinatorics (1998) 19, 718 On Free Deformations of the Braid Arrangement

Summary: Europ. J. Combinatorics (1998) 19, 7­18
On Free Deformations of the Braid Arrangement
We classify the hyperplane arrangements between the cones of the braid arrangement and the Shi
arrangement of type An-1 which are free, in the sense of Terao. We also prove that the cones of
the extended Shi arrangements of type An-1 are free, verifying part of a conjecture of Edelman and
c 1998 Academic Press Limited
There has been considerable interest in the past in analyzing specific families of hyperplane
arrangements from the perspective of freeness. Examples of such families have primarily in-
cluded classes of subarrangements of Coxeter arrangements. The subarrangements of the braid
arrangement An, the Weyl arrangement of type An-1, are known as the graphical arrange-
ments. They correspond naturally to graphs on n vertices. It follows mainly from the work of
Stanley [14] and is recorded in [5, §3] that free graphical arrangements correspond to chordal
graphs. Certain classes of arrangements between the root systems An-1 and Bn were studied
by J´ozefiak and Sagan [9]. These arrangements can also be related to graphs. Edelman and
Reiner [5] gave a complete classification of the free arrangements in this case and showed that
they correspond to threshold graphs. In a more recent work [6] these authors classified free
arrangements which arise as discriminantal arrangements of two-dimensional zonotopes with


Source: Athanasiadis, Christos - Department of Mathematics, University of Athens


Collections: Mathematics