$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions

We show that there are $f$-vectors of balanced simplicial complexes giving a source of simplicial complexes exhibiting a Boolean decomposition similar to a geometric Lefschetz decomposition. The objects we are working with are $h$-vectors of flag spheres and balanced simplicial complexes whose $f$-vectors are equal to them. This builds on work of Nevo--Petersen--Tenner on a conjecture of Nevo--Petersen that the gamma vector of an odd-dimensional flag sphere is the $f$-vector of a balanced simplicial complex (which was shown for barycentric subdivisions by Nevo--Petersen--Tenner). We can connect our decomposition to positivity questions on reciprocal/palindromic polynomials associated to flag spheres and geometric questions motivating them. In addition, we note that the degrees in the Lefschetz-like decomposition are not halved unlike the usual $h$-vector setting.