The γ-positivity of basic Eulerian polynomials via group actions

We provide combinatorial interpretation for the γ-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding γ-coefficients for derangements. Our results refine the classical γ  -positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brändén's modified Foata–Strehl action on permutations and the recent triple statistic (des,rix,aiddes,rix,aid) equidistributed with (exc,fix,majexc,fix,maj).