  The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive.
Define the positive Eulerian polynomial $A_n^+(t)$ as the polynomial obtained
when we sum descents over the alternating group. We show that $A_n^+(t)$ is
gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2$ (mod 4) we show
that $A_n^+(t)$ can be written as a sum of two gamma positive polynomials while
if $n \equiv 3$ (mod 4), we show that $A_n^+(t)$ can be written as a sum of
three gamma positive polynomials.
  Similar results are shown when we consider the positive type-D and type-D
Eulerian polynomials.
