Details:
Title  TypeB generalized triangulations and determinantal ideals  Author(s)  Daniel Soll, Volkmar Welker  Type  Article in Journal  Abstract  For n ≥ 3 , let Ω n be the set of line segments between the vertices of a convex n gon. For j ≥ 2 , a j crossing is a set of j line segments pairwise intersecting in the relative interior of the n gon. For k ≥ 1 , let Δ n , k be the simplicial complex of (typeA) generalized triangulations, i.e. the simplicial complex of subsets of Ω n not containing any ( k + 1 ) crossing. The complex Δ n , k has been the central object of many papers. Here we continue this work by considering the complex of typeB generalized triangulations. For this we identify line segments in Ω 2 n which can be transformed into each other by a 180∘rotation of the 2 n gon. Let F n be the set Ω 2 n after identification, then the complex D n , k of typeB generalized triangulations is the simplicial complex of subsets of F n not containing any ( k + 1 ) crossing in the above sense. For k = 1 , we have that D n , 1 is the simplicial complex of typeB triangulations of the 2 n gon as defined in [R. Simion, A typeB associahedron, Adv. Appl. Math. 30 (2003) 2–25] and decomposes into a join of an ( n − 1 ) simplex and the boundary of the n dimensional cyclohedron. We demonstrate that D n , k is a pure, k ( n − k ) − 1 + k n dimensional complex that decomposes into a k n − 1 simplex and a k ( n − k ) − 1 dimensional homologysphere. For k = n − 2 we show that this homologysphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of D n , k . On the algebraical side we give a term order on the monomials in the variables X i j , 1 ≤ i , j ≤ n , such that the corresponding initial ideal of the determinantal ideal generated by the ( k + 1 ) times ( k + 1 ) minors of the generic n × n matrix contains the Stanley–Reisner ideal of D n , k . We show that the minors form a GröbnerBasis whenever k ∈ 1 , n − 2 , n − 1 thereby proving the equality of both ideals and the unimodality of the h vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k < n .  Keywords  Triangulation, Associahedron, Cyclohedron, Determinantal ideal  ISSN  0012365X 
URL 
http://www.sciencedirect.com/science/article/pii/S0012365X08004366 
Language  English  Journal  Discrete Mathematics  Volume  309  Number  9  Pages  2782  2797  Year  2009  Edition  0  Translation 
No  Refereed 
No 
