Gröbner bases associated with positive roots and Catalan numbers
Tomonori Kitamura · Osaka City University (Osaka City University) · 2005
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 $T \ eq \\{ (1,2) , (1,3), \\ldots , (1,n) \\}$, 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 $T \ eq \\{ (1,2), (1,3), \\ldots , (1,n) \\}$ will be constructed. In particular, a new combinatorial expression of Catalan numbers arises.