55 research outputs found
Revlex-Initial 0/1-Polytopes
We introduce revlex-initial 0/1-polytopes as the convex hulls of
reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are
special knapsack-polytopes. It turns out that they have remarkable extremal
properties. In particular, we use these polytopes in order to prove that the
minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the
graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and
a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their
graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and
Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at
least one.Comment: Accepted for publication in J. Comb. Theory Ser. A; 24 pages;
simplified proof of Theorem 1; corrected and improved version of Theorem 4
(the average degree is now bounded by d+4 instead of d+8); several minor
corrections suggested by the referee
Revlex-initial 0/1-polytopes
AbstractWe introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers gnfac(d,n) of facets and the minimum average degree gavdeg(d,n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy gnfac(d,n)⩽3d and gavdeg(d,n)⩽d+4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one
Koszul homology and extremal properties of Gin and Lex
In a polynomial ring with variables, for every homogeneous ideal
and for every we consider the Koszul homology with
respect to a sequence of of generic linear forms and define the
Koszul-Betti number of to be the dimension of the
degree part of . In characteristic 0, we show that the
Koszul-Betti numbers of any ideal are bounded above by those of any gin of
and also by those of the Lex-segment of . We also investigate the set
of all the gin of and show that the Koszul-Betti numbers of any
ideal in are bounded below by those of the gin-revlex of and
present examples showing that in general there is no is such that
the Koszul-Betti numbers of any ideal in are bounded above by those
of .Comment: 21 pages, preprint 200
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 Petersen et
al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to
their result that the flag simplicial complex on ([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]×[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 the noncrossing complex, and the polytope
derived from it the dual Grassmann associahedron. We extend results of
Petersen et 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
isomorphismGk,n≅Gn−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 a Grassmann–Tamari order on 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 Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its
cyclically invariant part
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