Related papers: Shellable flag simplicial complexes of non-simple …
The flag f-vectors of three-colored complexes are characterized. This also characterizes the flag h-vectors of balanced Cohen-Macaulay complexes of dimension two, as well as the flag h-vectors of balanced shellable complexes of dimension…
We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal…
Shellability of a simplicial complex has many useful structural implications. In particular, it was shown by Danaraj and Klee that every shellable pseudo-manifold is a PL-sphere. The purpose of this paper is to prove the shellability of the…
In the focus of this paper is the operation of edge contraction. One can show that simplicial 3-polytope is flag iff contraction of any its edge gives simplicial 3-polytope. Our main result states that any flag simplicial 3-polytope can be…
A simplicial complex $\Delta$ is a virtually Cohen-Macaulay simplicial complex if its associated Stanley-Reisner ring $S$ has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length ${\rm codim}(S)$. We provide a…
Let $\D$ be a $(d-1)$-dimensional pure $f$-simplicial complex over vertex set $[n]$. In this paper, it is proved that $n=2d$ holds true if $\D$ is minimal Cohen-Macaulay. It is also indicated that the recent work of \cite{Dao2020} implies…
The $r$-fold edgewise subdivision is a well studied flag triangulation of the simplex with interesting algebraic, combinatorial and geometric properties. An important enumerative invariant, namely the local $h$-polynomial, of this…
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of…
The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the…
This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities,…
We introduce two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. Both complexes are intimately related to the bubble lattice introduced in our earlier article "Bubble Lattices I: Structure"…
We present a short proof of Reisner's Theorem, characterizing which simplicial complexes have a Cohen-Macaulay face ring. In some cases, we can also express some homological invariants of the face ring in terms of the reduced homology of…
One of the most common and effective methods of obtaining structural information on simplicial complexes is to use tools from algebraic geometry/commutative algebra (often motivated by properties of toric varieties). However, there is no…
We study the index of symmetry of a compact generalized flag manifold M=G/H endowed with an invariant Kaehler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces and we…
In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying…
We study combinatorial and algebraic properties of $t$-clique-free complexes, a family of simplicial complexes associated with finite simple graphs that generalize the classical independence complex. For a graph $G$ and an integer $t \ge…
Via the BGG correspondence a simplicial complex Delta on [n] is transformed into a complex of coherent sheaves on P^n-1. We show that this complex reduces to a coherent sheaf F exactly when the Alexander dual Delta^* is Cohen-Macaulay. We…
The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for $q$-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends…
Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus…
Suppose a group $G$ acts properly on a simplicial complex $\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\Gamma$ must be a subcomplex of…