Title: A generalized lower bound inequality and rigidity of a-balanced simplicial complexes
Abstract:
Generalized lower bound inequality asserts that the h-vector of any simplicial sphere is unimodal. Balanced GLBT asserts that stronger inequality holds for simplicial (d-1)-spheres having d-colorable 1-skeleton. To bridge between these two, we consider a-balanced simplicial spheres.
A pure simplicial complex is called (a_1,...,a_m)-balanced if vertices can be colored using m colors in such a way that every facet contains a_i vertices of the i-th color.
Generalizing (the inequality part of) GLBT, we give a tight lower bound of the ratio h_{k+1}/h_k when each a_i is at least 2k+1. The proof is done by adapting anisotropy proofs of Karu-Xiao to the setting of a-colored s.o.p.
I will also briefly discuss Fogelsanger's rigidity technique to go beyond simplicial spheres.