Related papers: Shellability and higher Cohen-Macaulay connectivit…

We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…

The (extended) Linial arrangement $\mathcal{L}_{\Phi}^m$ is a certain finite truncation of the affine Weyl arrangement of a root system $\Phi$ with a parameter $m$. Postnikov and Stanley conjectured that all roots of the characteristic…

A finitely generated module $M$ over a local ring is called a sequentially generalized Cohen-Macaulay module if there is a filtration of submodules of $M$: $M_0\subset M_1\subset ... \subset M_t=M$ such that $\dim M_0<\dim M_1< >... <\dim…

Topological Chern-Simons (CS) and BF theories and their holomorphic analogues are discussed in terms of de Rham and Dolbeault cohomologies. We show that Cech cohomology provides another useful description of the above topological and…

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called…

The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of…

In this article, we generalize a previously defined set of axioms for a closure operation that induces balanced big Cohen-Macaulay modules. While the original axioms were only defined in terms of finitely generated modules, these new ones…

We introduce the notion of a strong generalized holomorphic (SGH) fiber bundle and develop connection and curvature theory for an SGH principal $G$-bundle over a regular generalized complex (GC) manifold, where $G$ is a complex Lie group.…

We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $\Pi_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $\Delta_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that…

We prove that the complex of proper non-trivial non-degenerate subspaces of a finite-dimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same…

In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials.…

We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if…

For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting…

Associated to a simple undirected graph $G$ is a simplicial complex $\Delta_G$ whose faces correspond to the independent sets of $G$. A graph $G$ is called vertex decomposable if $\Delta_G$ is a vertex decomposable simplicial complex. We…

We discuss the structure of the set $\Delta$ consisting of pairs of closed subspaces that have a common complement in a Hilbert space previously studied by Lauzon and Treil (J. Funct. Anal. 212: 500--512, 2004). We prove that $\Delta$ is…

We construct scattering diagrams for Chekhov-Shapiro's generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out…