On algebraic and combinatorial properties of weighted simplicial complexes

TL;DR

This paper introduces a novel polarization approach to analyze weighted simplicial complexes, linking their properties to unweighted complexes and mixed wreath products, with implications for algebraic and combinatorial invariants.

Contribution

The paper proposes a new polarization method for WSCs that preserves key properties and invariants, connecting weighted complexes to unweighted structures and mixed wreath products.

Findings

Polarization constructs unweighted complexes from WSCs.

Preservation of properties like shellability and vertex-decomposability.

Analysis of algebraic invariants such as Betti numbers and associated primes.

Abstract

Weighted simplicial complexes (WSCs) are powerful tools for describing weighted cloud data or networks with weighted nodes. In this paper, we propose a novel approach to study WSCs via the concept of polarization. Polarization of a WSC allows one to construct a new (unweighted) simplicial complex which coincides with an object called the mixed wreath product. This new construction preserves several properties and invariants of the underlying simplicial complex of a WSC. Our main focus is to analyze WSCs through their underlying simplicial complexes and mixed wreath products. Combinatorially, we investigate properties such as vertex-decomposability, shellability, constructibility; algebraically, we study Betti numbers, associated primes and primary decompositions of ideals associated to WSCs.