Toric rings associated with root systems and conic divisorial ideals via matroid theory

Abstract

We study conic divisorial ideals from the viewpoint of matroid theory and apply the resulting framework to toric rings arising from root systems. For a toric ring, we describe the polytope representing divisor classes corresponding to conic divisorial ideals in terms of matroids. We then turn to the toric ring $R_P$ associated with a certain subset $P$ of a classical root system, called a signed poset. We compute the divisor class group and characterize the ($\mathbb{Q}$-)Gorenstein property of $R_P$ in terms of $P$. Moreover, we also construct a polytope characterizing the conic divisorial ideals of $R_P$. This recovers and extends previous results on Hibi rings to our toric rings.