Quadratic initial ideals of root systems (Q2782617)

Let \({\mathcal A}\) be a configuration in \(\mathbb{R}^n\), i.e. a finite subset of \(\mathbb{Z}^n\), let \(K\) be a field and let \({\mathcal R}_{K}[{\mathcal A}]\) be the \(K\)-subalgebra of the Laurent polynomial ring \(K[t_1, t^{-1}_1, \ldots, t_n, t^{-1}_n, s]\) generated by the monomials \(t_{1}^{a_{1}}. \cdots . t_{n}^{a_{n}} s\), for all \((a_{1}, \ldots , a_{n}) \in {\mathcal A}\). Let \(K[{\mathcal A}]\) be the polynomial ring in the variables \(x_{(a_1, \ldots , a_n)}\) for all \((a_{1}, \ldots , a_{n}) \in {\mathcal A}\); the \textit{toric ideal} \(I_{\mathcal A}\) of \({\mathcal A}\) is the kernel of the (surjective) morphism \(\pi : K[{\mathcal A}] \rightarrow {\mathcal R}_K[{\mathcal A}]\) defined by \(x_{(a_1, \ldots , a_n)} \mapsto t_{1}^{a_{1}}. \cdots . t_{n}^{a_{n}} s\). NEWLINENEWLINENEWLINEA root system is a finite set of vectors in a Euclidean space with certain properties [see \textit{W. Fulton} and \textit{J. Harris}, ``Representation theory. A first course'' (1991; Zbl 0744.22001), p. 320], and it is an important tool in the classification of semisimple Lie algebras. The root systems associated with the classical Lie algebras are usually denoted by \({\text A}_{n -1}\), \(\text{B}_n\), \(\text{C}_n\) and \(\text{D}_n\). Taking only the so called positive roots in each system, and adding the origin \(\text{O}\) of \(\mathbb{R}^n\) to the subsets thus obtained, we get the sets: NEWLINENEWLINENEWLINE\({\text{A}}_{n -1}^{(+)} = \{\text{O}\} \cup \{\text{e}_i - \text{e}_j\); \(1 \leq i < j \leq n\}\), NEWLINENEWLINENEWLINE\(\text{B}_n^{(+)} = {\text{A}}_{n -1}^{(+)} \cup \{ \text{e}_1, \ldots , \text{e}_n\} \cup \{ \text{e}_i + \text{e}_j\); \(1 \leq i < j \leq n\}\), NEWLINENEWLINENEWLINE\(\text{C}_n^{(+)} = {\text A}_{n -1}^{(+)} \cup \{2 \text{e}_1, \ldots , 2 \text{e}_n\} \cup \{ \text{e}_i + \text{e}_j\); \(1 \leq i < j \leq n\}\), NEWLINENEWLINENEWLINE\(\text{D}_n^{(+)} = {\text A}_{n -1}^{(+)} \cup \{ \text{e}_i + \text{e}_j\); \(1 \leq i < j \leq n\}\) NEWLINENEWLINENEWLINEwhere \(\text{e}_i\) is the \(i\)-th coordinate vector of \(\mathbb{R}^n\). We may take these sets to be configurations, if we associate to each of its vectors their coordinates in \(\mathbb{R}^n\). \textit{I. M. Gelfand, M. T. Graev} and \textit{A. Postnikov} [in: The Arnold-Gelfand mathematical seminars: geometry and singularity theory, 205-221 (1997; Zbl 0876.33011)], using a determined reverse lexicographic monomial order, found a Gröbner basis for the toric ideal \(I_{{\text A}_{n -1}^{(+)}}\) formed by binomials whose monomials are quadratic and squarefree. NEWLINENEWLINENEWLINEIn the paper under review, the authors show that the same is true for the toric ideals \(I_{\text{B}_n^{(+)}}\), \(I_{\text{C}_n^{(+)}}\) and \(I_{\text{D}_n^{(+)}}\). They determine explicitly such a basis for each of these ideals, and note that, as a consequence of their result, the convex polytope of the convex hull of either \(\text{B}_n^{(+)}\), \(\text{C}_n^{(+)}\) or \(\text{D}_n^{(+)}\) possesses a regular unimodular triangularization arising from a flag complex; also, the affine semigroup rings \({\mathcal R}_K[\text{B}_n^{(+)}]\), \({\mathcal R}_K[\text{C}_n^{(+)}]\) and \({\mathcal R}_K[\text{D}_n^{(+)}]\) are Koszul.