Toric ideals associated with root systems are studied.
First, we show the existence of squarefree quadratic initial
ideals for configurations arising from the set of positive roots
of root systems together with the origin. In particular, corresponding
affine semigroup rings are Koszul. Second, we discuss squarefree
initial ideals and unimodular coverings for configurations arising from the set of positive roots of root systems together with the
origin, without the origin, and thier subconfigurations.
It is well-known
that corresponding affine semigroup ring is normal if the toric
ideal has a squarefree initial ideal or if the convex
hull of the configuration has a unimodular covering.