Shifted simplicial complexes are Laplacian integral

Shifted Simplicial Complexes Are Laplacian Integral

We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial...

Matroids vs. Shifted simplicial complexes

The families of matroid independence complexes and pure shifted simplicial complexes are two remarkable classes of simplicial complexes that share several structural properties, but for which theorems are typically proved in different ways. In a groundbreaking paper Kook, Reiner and Stanton [4] proved that matroid complexes are Laplacian integral. Afterwards Duval and Reiner [1] proved the anal...

Integral Homology of Random Simplicial Complexes

On the Betti Numbers of Shifted Complexes of Stable Simplicial Complexes

Let ∆ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex ∆s of ∆ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, . . . , xn] of the symmetric algebraic shifted, exterior algebraic shifted and combinatorial shifted complexes of ∆ are equal.

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...