Chow and augmented Chow polynomials as evaluations of Poincar\'e-extended ab-indices

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labellings, are obtained as evaluations of their Poincar\'e-extended ab-indices. This implies in particular explicit combinatorial gamma-positive expansions for both.