80 research outputs found
Erdos-Ko-Rado theorems for simplicial complexes
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to
Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in
terms of the graph's independent sets. Since the family of all independent sets
of a graph forms a simplicial complex, it is natural to further generalize the
Erdos-Ko-Rado property to an arbitrary simplicial complex. An advantage of
working in simplicial complexes is the availability of algebraic shifting, a
powerful shifting (compression) technique, which we use to verify a conjecture
of Holroyd and Talbot in the case of sequentially Cohen-Macaulay near-cones.Comment: 14 pages; v2 has minor changes; v3 has further minor changes for
publicatio
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
Linear resolutions of powers and products
The goal of this paper is to present examples of families of homogeneous
ideals in the polynomial ring over a field that satisfy the following
condition: every product of ideals of the family has a linear free resolution.
As we will see, this condition is strongly correlated to good primary
decompositions of the products and good homological and arithmetical properties
of the associated multi-Rees algebras. The following families will be discussed
in detail: polymatroidal ideals, ideals generated by linear forms and Borel
fixed ideals of maximal minors. The main tools are Gr\"obner bases and Sagbi
deformation
Toric complexes and Artin kernels
A simplicial complex L on n vertices determines a subcomplex T_L of the
n-torus, with fundamental group the right-angled Artin group G_L. Given an
epimorphism \chi\colon G_L\to \Z, let T_L^\chi be the corresponding cover, with
fundamental group the Artin kernel N_\chi. We compute the cohomology jumping
loci of the toric complex T_L, as well as the homology groups of T_L^\chi with
coefficients in a field \k, viewed as modules over the group algebra \k\Z. We
give combinatorial conditions for H_{\le r}(T_L^\chi;\k) to have trivial
\Z-action, allowing us to compute the truncated cohomology ring, H^{\le
r}(T_L^\chi;\k). We also determine several Lie algebras associated to Artin
kernels, under certain triviality assumptions on the monodromy \Z-action, and
establish the 1-formality of these (not necessarily finitely presentable)
groups.Comment: 34 page
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