Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes

Abstract

A colored complex of type $\mathbf{a}=(a_1,\dots ,a_n)$ is a simplicial complex $\Delta$ on a vertex set $V$, together with an ordered partition $(V_1,\dots ,V_n)$ of $V$, such that every face $F$ of $\Delta$ satisfies $|F\cap V_i|\le a_i$. For each $\mathbf{b}=(b_1,\dots ,b_n)\le \mathbf{a}$, let $f_{\mathbf{b}}$ be the number of faces $F$ of $\Delta$ such that $|F\cap V_i|=b_i$. The array of integers $\{f_{\mathbf{b}}{\}}_{\mathbf{b}\le \mathbf{a}}$ is called the fine $f$-vector of $\Delta$, and it is a refinement of the $f$-vector of $\Delta$. In this paper, we generalize the notion of Macaulay representations and give a numerical characterization of the fine $f$-vectors of colored complexes of arbitrary type, in terms of these generalized Macaulay representations. As part of the proof, we introduce the property of $\mathbf{a}$-Macaulay decomposability for simplicial complexes, which implies vertex-decomposability, and we show that every pure color-shifted balanced complex $\Delta$ of type $\mathbf{a}$ is $\mathbf{a}$-Macaulay decomposable. Combined with previously known results, we also obtain a numerical characterization of the flag $f$-vectors of completely balanced Cohen-Macaulay complexes.

arXiv comment28 pages, 6 figures; v2: significantly reduced page length, revised introduction

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