37 research outputs found
Exponential formulas for models of complex reflection groups
In this paper we find some exponential formulas for the Betti numbers of the
De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the
complex reflection groups G(r,p,n). Our formulas are different from the ones
already known in the literature: they are obtained by a new combinatorial
encoding of the elements of a basis of the cohomology by means of set
partitions with weights and exponents.
We also point out that a similar combinatorial encoding can be used to
describe the faces of the real spherical wonderful models of type
A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential
formulas for the f-vectors of the associated nestohedra: the Stasheff's
associahedra (in this case closed formulas are well known) and the graph
associahedra of type D_n.Comment: with respect to v.1: misprint corrected in Example 3.
Note on resonance varieties
We study the irreducibility of resonance varieties of graded rings over an
exterior algebra E with particular attention to Orlik-Solomon algebras. We
prove that for a stable monomial ideal in E the first resonance variety is
irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane
arrangement, then we show that its first resonance variety is irreducible if
and only if the subideal of J generated by all degree 2 elements has a 2-linear
resolution. As an application we characterize those hyperplane arrangements of
rank less than or equal to 3 where J is componentwise linear. Higher resonance
varieties are also considered. We prove results supporting a conjecture of
Schenck-Suciu relating the Betti numbers of the linear strand of J and its
first resonance variety. A counter example is constructed that this conjecture
is not true for arbitrary graded ideals
DE CONCINI AND PROCESI MODELS OF REFLECTION GROUPS AND COXETER GROUPS
Study of De Concini and Procesi Wonderful models for subspace arrangement related to subspace arrangement generated by reflection groups and Coxeter groups
On the inclusion ideal graph of a poset
Let (P,≤) be an atomic partially ordered set (poset, briefly) with a minimum element 0 and \u1d57f(P) the set of nontrivial ideals of P. The inclusion ideal graph of P, denoted by Ω(P), is an undirected and simple graph with the vertex set \u1d57f(P) and two distinct vertices I, J ∈ \u1d57f(P) are adjacent in Ω(P) if and only if I ⊂ J or J ⊂ I. We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that Ω(P) is not connected if and only if P = {0, a1, a2}, where a1, a2 are two atoms. Moreover, it is shown that if Ω(P) is connected, then diam(Ω(P)) ≤ 3. Also, we show that if Ω(P) contains a cycle, then girth(Ω(P)) ∈ {3, 6}. Furthermore, all posets based on their diameters and girths of inclusion ideal graphs are characterized. Among other results, all posets whose inclusion ideal graphs are path, cycle and star are characterized