Compressed polytopes, initial ideals and complete multipartite graphs
type: Item
instance of: scholarly article
title: Compressed polytopes, initial ideals and complete multipartite graphs
zbMATH Open document ID: 0943.13016
author: Hidefumi Ohsugi
author: Takayuki Hibi
published in: Illinois Journal of Mathematics
publication date: 2000-09-11T00:00:00Z
review text: The second hypersimplex of order \(d\) is the convex polytope \(\Delta(2,d)\) which is the convex hull of the configuration \({\mathcal A}_d= \{{\mathbf e}_i+ {\mathbf e}_j\); \(1\leq i< j\leq d\}\) in \(\mathbb{R}^d\), where \({\mathbf e}_i\) is the \(i\)-th unit coordinate vector of \(\mathbb{R}^d\). If \(G\) is a finite connected graph having no loop and no multiple edge on the vertex set \([d]= \{1, 2,\dots, d\}\) with the edge set \(E(G)\), then we write \({\mathcal A}_G\) for the subset \(\{{\mathbf e}_i+ {\mathbf e}_j\in{\mathcal A}_d\); \(\{i,j\}\in E(G)\}\) of \({\mathcal A}_d\). The edge polytope \({\mathcal P}_G\) of \(G\) is the convex hull of \({\mathcal A}_G\) in \(\mathbb{R}^d\). Let \(K[t_1, t_2,\dots, t_d]\) denote the polynomial ring in \(d\) variables over a field \(K\). The affine semigroup ring \(K[G]\) which is generated by all monomials \(t_i t_j\) with \(\{i,j\}\in E(G)\) is called the edge ring of \(G\). When \(G\) is the complete graph on \([d]\), its edge polytope is the second hypersimplex of order \(d\) and its edge ring is the second squarefree Veronese subring of \(K [t_1, t_2,\dots, t_d]\). In the present paper the authors are interested in edge polytopes and edge rings of complete multipartite graphs on \([d]\). Here, a complete multipartite graph on \([d]\) is a finite graph on \([d]\) such that, for a suitable decomposition \([d]= V_1\cup V_2\cup \dots\cup V_n\) of \([d]\), its edge set consists of all \(\{k, \ell\}\) with \(k\in V_i\) and \(\ell\in V_j\) for some \(i\neq j\). First, in section 1, the notion of the algebra of Segre-Veronese type which generalizes both Segre products and Veronese subrings of polynomial rings will be presented. Such algebras are affine semigroup rings which possess squarefree quadratic initial ideals; in particular, these algebras are normal, Cohen-Macaulay and Koszul. The edge ring \(K[G]\) of a finite connected graph \(G\) is an algebra of Segre-Veronese type if and only if \(G\) is a complete mult…
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 05C62
zbMATH DE Number: 1448998
zbMATH Keywords: compressed polytope
zbMATH Keywords: edge polytopes
zbMATH Keywords: edge rings of complete multipartite graphs
zbMATH Keywords: \(f\)-vector
zbMATH Keywords: Ehrhart polynomial
zbMATH Keywords: Stanley-Reisner ideal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1448998