On f- and h- vectors of relative simplicial complexes

TL;DR

This paper characterizes the possible f- and h-vectors of relative simplicial complexes, linking combinatorial structures with algebraic properties like Cohen--Macaulayness, and resolves a longstanding question in the field.

Contribution

It provides the first comprehensive characterization of f- and h-vectors of relative complexes, connecting combinatorial and algebraic invariants and addressing open problems.

Findings

Characterization of f-vectors of relative complexes on fixed ground set size.

Characterization of h-vectors of Cohen--Macaulay relative complexes.

Resolution of Björner's question on minimal faces.

Abstract

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates. Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relative complexes as well as $h$-vectors of Cohen--Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Bj\"orner.