2,296 research outputs found
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Subword complexes in Coxeter groups
Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma
and an element in \Pi determine a {\em subword complex}, as introduced in our
paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword
complexes are demonstrated here to be homeomorphic to balls or spheres, and
their Hilbert series are shown to reflect combinatorial properties of reduced
expressions in Coxeter groups. Two formulae for double Grothendieck
polynomials, one of which is due to Fomin and Kirillov, are recovered in the
context of simplicial topology for subword complexes. Some open questions
related to subword complexes are presented.Comment: 14 pages. Final version, to appear in Advances in Mathematics. This
paper was split off from math.AG/0110058v2, whose version 3 is now shorte
Koszul algebras and regularity
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford
regularity. We describe several techniques to establish the Koszulness of
algebras. We discuss variants of the Koszul property such as strongly Koszul,
absolutely Koszul and universally Koszul. We present several open problems
related with these notions and their local variants
Weighted -cohomology of Coxeter groups
Given a Coxeter system and a positive real multiparameter \bq, we
study the "weighted -cohomology groups," of a certain simplicial complex
associated to . These cohomology groups are Hilbert spaces, as
well as modules over the Hecke algebra associated to and the
multiparameter . They have a "von Neumann dimension" with respect to the
associated "Hecke - von Neumann algebra," . The dimension of the
cohomology group is denoted . It is a nonnegative real number
which varies continuously with . When is integral, the
are the usual -Betti numbers of buildings of type and thickness
. For a certain range of , we calculate these cohomology groups as
modules over and obtain explicit formulas for the . The
range of for which our calculations are valid depends on the region of
convergence of the growth series of . Within this range, we also prove a
Decomposition Theorem for , analogous to a theorem of L. Solomon on the
decomposition of the group algebra of a finite Coxeter group.Comment: minor change
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
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