On subdivisions of simplicial complexes: Characterizing local \(h\)- vectors (Q1329193)

Given a finite \(d\)-dimensional simplicial complex \(\Delta\), the \(f\)- vector is the \((d+ 1)\)-tuple \((f_ 0,\dots,f_ d)\), where \(f_ i\) is the number of \(i\)-dimensional faces of \(\Delta\). By convention, \(f_{- 1}= 1\) unless \(\Delta= \varnothing\), then \(f_{-1}= 0\). An algebraically desirable alternative to the \(f\)-vector is the \(h\)-vector \((h_ 0,\dots,h_ d)\), which is related to the \(f\)-vector by \[ \sum^ d_{i=0} f_{i-1}(x- i)^{d-i}= \sum^ d_{i=0} h_ i x^{d-i}. \] Again related is the local \(h\)-vector. The problem resolved in this paper is a complete characterization of local \(h\)-vectors of subdivisions: if \(l= (l_ 0,\dots,l_ d)\in \mathbb{Z}^{d+1}\), then \(l= l_ V(\Gamma)\) for some subdivision \(\Gamma\) of \(2^ V\) if and only if \(l\) is symmetric \((l_ i= l_{d-i}\) for \(0\leq i\leq d)\) and \(l_ 0= 0\) and \(l_ 1\geq 0\). A complete characterization of local \(h\)-vectors of regular subdivisions is also given.