On $\g$- and local $\g$-Vectors of the Interval Subdivision

TL;DR

This paper proves the nonnegativity of $ ext{g}$- and local $ ext{g}$-vectors for interval subdivisions of simplicial complexes, linking these vectors to balanced complexes and answering an open question.

Contribution

It establishes the nonnegativity of $ ext{g}$- and local $ ext{g}$-vectors in specific subdivisions, connecting them to balanced complexes and resolving an open problem.

Findings

$ ext{g}$-vector of interval subdivision is nonnegative

Local $ ext{g}$-vector of a simplex is nonnegative

Such $ ext{g}$-vector corresponds to an $f$-vector of a balanced complex

Abstract

We show that the $\g$-vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric $h$-vector is nonnegative. In particular, we prove that such $\g$-vector is the $f$-vector of some balanced simplicial complex. Moreover, we show that the local $\g$-vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al.