Gröbner bases associated with positive roots and Catalan numbers

Abstract

Let $\mathbf{A}_{n-1}^+ \subset \mathbb{Z}^n$ denote the set of positive roots of the root system $\mathbf{A}_{n-1}$ and $I_{\mathbf{A}_{n-1}^+}$ its toric ideal. The purpose of the present paper is to study combinatorics and algebra on $\mathbf{A}_{n-1}^+$ and $I_{\mathbf{A}_{n-1}^+}$. First, it will be proved that $I_{\mathbf{A}_{n-1}^+}$ induces an initial ideal $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$ which is generated by quadratic squarefree monomials together with cubic squarefree monomials. Second, we will associate each maximal face $\sigma$ of the unimodular triangulation $\Delta$ arising from $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$ with a certain subgraph $G_\sigma$ on $[n] = \{1,\ldots,n\}$. Third, noting that the number of maximal faces of $\Delta$ is equal to that of anti-standard trees $T$ on $[n]$ with $T \neq \{ (1,2) , (1,3), \ldots , (1,n) \}$, an explicit bijection between the set $\{ G_\sigma \colon \sigma\ \text{is a maximal face of}\ \Delta \}$ and that of anti-standard trees $T$ on $[n]$ with $T \neq \{ (1,2), (1,3), \ldots , (1,n) \}$ will be constructed. In particular, a new combinatorial expression of Catalan numbers arises.