Balanced squeezed Complexes

TL;DR

This paper introduces balanced squeezed complexes derived from color-squarefree monomials, demonstrating their combinatorial and algebraic properties, including vertex-decomposability, explicit Stanley-Reisner ideals, and Betti number computations, generalizing classical stable ideals.

Contribution

The authors define balanced squeezed complexes from order ideals, analyze their properties, and connect their algebraic invariants to combinatorial structures, extending the theory of squeezed complexes and stable monomial ideals.

Findings

Balanced squeezed complexes are vertex-decomposable.

Flag h-vectors are determined by the underlying order ideal.

Explicit Stanley-Reisner ideals are provided for these complexes.

Abstract

Given any order ideal $U$ consisting of color-squarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional simplicial complex $\Delta_{\mathrm{bal}}(U)$ that we call a balanced squeezed complex. In fact, these complexes have properties similar to squeezed balls as introduced by Kalai and the more general squeezed complexes, introduced by the authors. We show that any balanced squeezed complex is vertex-decomposable and that its flag $h$-vector can be read off from the underlying order ideal. Moreover, we describe explicitly its Stanley-Reisner ideal $I_{\Delta_{\mathrm{bal}}(U)}$. If $U$ is also shifted, we determine the multigraded generic initial ideal of $I_{\Delta_{\mathrm{bal}}(U)}$ and establish that the balanced squeezed complex $\Delta_{\mathrm{bal}}(U)$ has the same graded Betti numbers as the complex obtained from color-shifting it. We also introduce a class of color-squarefree monomial ideals that may be viewed as a generalization of the classical squarefree stable monomial ideals and show that their graded Betti numbers can be read off from their minimal generators. Moreover, we develop some tools for computing graded Betti numbers.