Generalizing the concepts of Stanley–Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in Rd the Zd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen–Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Zd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences...

The torus manifolds have been defined and studied by Masuda and Panov ([7]) who in particular also describe its cohomology ring structure. In this note we shall describe the topological K-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a homology poytope whose nerve is shellable) in terms of generators and relations. Since these torus...

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W . We study the partially ordered set of B×Borbits (with respect to Zariski closure inclusion) within a G × G-orbit of M . This is the same as studying a W ×W -or...

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal ...

We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of LasVergnas’s external/internal order <ext/int on M provides a shelling of Act<(M). We also show that every linear extension of LasVergnas’s internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same hvecto...

Anders Björner characterized which finite graded partially ordered sets arise as the closure relation on cells of a finite regular CW complex. His characterization of these “CW posets” required each open interval (0̂, u) to have order complex homeomorphic to a sphere of dimension rk(u)− 2. Work of Danaraj and Klee showed that sufficient conditions were for the poset to be thin and shellable. The...

Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex ∆mW in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆ W is isomorphic to the cluster complex corresponding to W , defined in [9]. We enumerate the faces of ∆ W and show that the entries of its h-vector are given by the generalized Narayana numbers N W (i), de...

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S = k[x1, x2, ..., xn]; this includes the case of powers of the homogeneous maximal ideal (x1, x2, ..., xn) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in...

Abstract An h -tiling on a finite simplicial complex is partition of its geometric realization by maximal simplices deprived several codimension one faces together with possibly their remaining face highest codimension. In this last case, the tiles are said to be critical. thus induces partitioning poset closed or semi-open intervals. We prove existence -tilings every after finitely many stella...