Name
Title | ||
|---|---|---|
Normalized volumes of configurations related with root systems and complete bipartite graphs |
Abstract | ||
|---|---|---|
Let Φ ⊂ Zn denote one of the classical irreducible root systems An-1, Bn, Cn, and Dn, and write Φ(+) for the configuration consisting of all positive roots of Φ together with the origin of Rn. Gelfand, Graev and Postnikov, in: V.I. Arnold, I.M. Gelfand, M. Smirnov, V.S. Retakh (Eds.), Arnold-Gelfand, Mathematics Seminars, Geometry and Singularity Theory, Birkhäuser, Boston, 1997, pp. 205-221 showed that by constructing an explicit unimodular triangulation, the normalized volume of the convex hull of An-1(+) is equal to the Catalan number. On the other hand, Fong (Triangulations and Combinatorial Properties of Convex Polytopes, Dissertation, MIT Press, Cambridge, MA, 2000) computed the normalized volume of the convex hull of each of the configurations Bn(+), Cn(+) and Dn(+). Moreover, the normalized volume of the convex hull of the subconfiguration of An-1(+) arising from a complete bipartite graph was computed by Ohsugi and Hibi (Illinois J. Math. 44 (2000) 391) and Fong. The purpose of the present paper is, via the theory of Gröbner bases of toric ideals and triangulations, to compute the normalized volume of the convex hull of each of the subconfigurations of Bn(+), Cn(+) and Dn(+) arising from a complete bipartite graph. |
Year | DOI | Venue |
|---|---|---|
2003 | 10.1016/S0012-365X(02)00690-8 | Discrete Mathematics |
Keywords | Field | DocType |
convex hull,m. gelfand,initial ideals,m. smirnov,catalan number,combinatorial properties,normalized volume,complete bipartite graph,unimodular triangulations,convex polytopes,illinois j. math,root system,configurations bn,complete bipartite graphs,root systems,convex polytope | Discrete mathematics,Complete graph,Complete bipartite graph,Combinatorics,Catalan number,Bipartite graph,Convex hull,Triangulation (social science),Polytope,Unimodular matrix,Mathematics | Journal |
Volume | Issue | ISSN |
268 | 1-3 | Discrete Mathematics |
Citations | PageRank | References |
1 | 0.51 | 0 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Hidefumi Ohsugi | 1 | 27 | 10.42 |
| Takayuki Hibi | 2 | 94 | 30.08 |