The $f$- and $h$-vectors of Interval Subdivisions

The interval subdivision Int$(\Delta)$ of a simplicial complex $\Delta$ was introduced by Walker. We give the complete combinatorial description of the entries of the transformation matrices from the $f$- and $h$-vectors of $\Delta$ to the $f$- and $h$-vectors of Int$(\Delta)$. We show that if $\Delta$ has non-negative $h$-vector then the $h$-polynomial of its interval subdivision has only real roots. As a consequence, we prove the Charney-Davis conjecture for Int$(\Delta)$, if $\Delta$ has non-negative reciprocal $h$-vector.