NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON

We study a natural generalization of the noncrossing relation between pairs of elements in$[n]$to$k$-tuples in$[n]$that was first considered by Petersenet al.[J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on$\binom{[n]}{k}$induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product$[k]\times [n-k]$of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for$k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex thenoncrossing complex, and the polytope derived from it the dualGrassmann associahedron. We extend results of Petersenet al.[J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define aGrassmann–Tamari orderon maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersenet al.[J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.

Noncrossing sets and a Graßmannian associahedron

International audience We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc. Nous étudions une généralisation naturelle de la relation entre les paires d’éléments non-croisés de $[n]$ et les $k$-uplets de $[n]$. Nous montrons que le complexe simplicial de drapeau sur $\binom{[n]}{k}$ induit par cette relation est une triangulation régulière, unimodulaire et de drapeau du polytope d’ordre de l’ensemble partiellement ordonné obtenu par le produit $[k] \times [n-k]$ des deux chaînes, et c’est la jointure d’un simplexe et une sphère (c’est-à-dire qu’elle est une triangulation de Gorenstein). Cela montre l’existence d’un polytope simplicial de drapeau dont l’idéal de Stanley-Reisner est un idéal initial de l’idéal de Graßmann-Plücker, tandis que les constructions précédentes d’un tel polytope ne garantissaient pas la propriété de drapeau. Le complexe simplicial et le polytope qui en découle reflètent naturellement les relations entre les Grassmanniens avec différents paramètres, en particulier l’isomorphisme $G_{k,n} \cong G_{n-k,n}$. Ce complexe simplicial est étroitement lié au complexe de séparabilité faible étudié par Zelevinskyet Leclerc.

The Mean Width of Circumscribed Random Polytopes

For a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

A convex 3-complex not simplicially isomorphic to a strictly convex complex

A set X in euclidean space is convex if the line segment joining any two points of X is in X. If X is convex, every boundary point is on an (n − 1)-plane which contains X in one of its two closed half-spaces. Such a plane is called a support plane for X. A simplicial complex K in is called strictly convex if |K| (the underlying space of K) is convex and if, for every simplex σ in ∂K (the boundary of K) there is a support plane for |K| whose intersection with |K| is precisely σ In this case |K| is often called a simplicial polytope.