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arXiv · 2605.02532

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

Koji Matsushita
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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.

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Koji Matsushita. 2026-05-04. Toric rings associated with root systems and conic divisorial ideals via matroid theory. https://arxiv.org/abs/2605.02532

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