Balanced squeezed Complexes
Abstract
Given any order ideal $U$ consisting of colorsquarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d1)$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 vertexdecomposable and that its flag $h$vector can be read off from the underlying order ideal. Moreover, we describe explicitly its StanleyReisner 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 colorshifting it. We also introduce a class of colorsquarefree 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.01521
 Bibcode:
 2020arXiv200701521J
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Commutative Algebra;
 O5E45;
 13F55;
 13D02
 EPrint:
 22 pages