Quadratic initial ideals of root systems

Abstract:

Let $\Phi \subset \mathbb {Z}^{n}$ be one of the root systems $\mathbf {A}_{n-1}$, $\mathbf {B}_n$, $\mathbf {C}_n$ and $\mathbf {D}_n$ and write $\Phi ^{(+)}$ for the set of positive roots of $\Phi$ together with the origin of $\mathbb {R}^{n}$. Let $K[\mathbf {t}, \mathbf {t}^{-1}, s]$ denote the Laurent polynomial ring $K[t_1, t_1^{-1}, \ldots , t_n, t_n^{-1}, s]$ over a field $K$ and write $\mathcal {R}_K[\Phi ^{(+)}]$ for the affine semigroup ring which is generated by those monomials $\mathbf {t}^{\mathbf {a}} s$ with $\mathbf {a}\in \Phi ^{(+)}$, where $\mathbf {t}^{\mathbf {a}} = t_1^{a_1} \cdots t_n^{a_n}$ if $\mathbf {a}= (a_1, \ldots , a_n)$. Let $K[\Phi ^{(+)}] = K[\{x_{\mathbf {a}} ; \mathbf {a}\in \Phi ^{(+)} \}]$ denote the polynomial ring over $K$ and write $I_{\Phi ^{(+)}}$ $( \subset K[\Phi ^{(+)}] )$ for the toric ideal of $\Phi ^{(+)}$. Thus $I_{\Phi ^{(+)}}$ is the kernel of the surjective homomorphism $\pi : K[\Phi ^{(+)}] \to \mathcal {R}_{K}[\Phi ^{(+)}]$ defined by setting $\pi (x_{\mathbf {a}}) = \mathbf {t}^{\mathbf {a}} s$ for all $\mathbf {a}\in \Phi ^{(+)}$. In their combinatorial study of hypergeometric functions associated with root systems, Gelfand, Graev and Postnikov discovered a quadratic initial ideal of the toric ideal $I_{\mathbf {A}_{n-1}^{(+)}}$ of $\mathbf {A}_{n-1}^{(+)}$. The purpose of the present paper is to show the existence of a reverse lexicographic (squarefree) quadratic initial ideal of the toric ideal of each of $\mathbf {B}_n^{(+)}$, $\mathbf {C}_n^{(+)}$ and $\mathbf {D}_n^{(+)}$. It then follows that the convex polytope of the convex hull of each of $\mathbf {B}_n^{(+)}$, $\mathbf {C}_n^{(+)}$ and $\mathbf {D}_n^{(+)}$ possesses a regular unimodular triangulation arising from a flag complex, and that each of the affine semigroup rings $\mathcal {R}_K[\mathbf {B}_n^{(+)}]$, $\mathcal {R}_K[\mathbf {C}_n^{(+)}]$ and $\mathcal {R}_K[\mathbf {D}_n^{(+)}]$ is Koszul.