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Quadratic initial ideals of root systems


Authors: Hidefumi Ohsugi and Takayuki Hibi
Journal: Proc. Amer. Math. Soc. 130 (2002), 1913-1922
MSC (2000): Primary 13P10
DOI: https://doi.org/10.1090/S0002-9939-01-06411-5
Published electronically: December 27, 2001
MathSciNet review: 1896022
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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.


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Additional Information

Hidefumi Ohsugi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan
Email: ohsugi@math.sci.osaka-u.ac.jp

Takayuki Hibi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan
Email: hibi@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-01-06411-5
Received by editor(s): August 8, 2000
Received by editor(s) in revised form: January 29, 2001
Published electronically: December 27, 2001
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society

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