The commutant of divided difference operators, Klyachko's genus, and the comaj statistic

In [Hamaker-Pechenik-Speyer-Weigandt, Nenashev, Pechenik-Weigandt] are studied certain operators on polynomials and power series that commute with all divided difference operators $\partial_i$. We introduce a second set of "martial" operators {\martial_i} that generate the full commutant, and show how a Hopf-algebraic approach naturally reproduces the operators $\xi^\nu$ from [Nenashev]. We then pause to study Klyachko's homomorphism $H^*(Fl(n)) \to H^*($the permutahedral toric variety$)$, and extract the part of it relevant to Schubert calculus, the "affine-linear genus''. This genus is then re-obtained using Leibniz combinations of the {\martial_i}. We use Nadeau-Tewari's $q$-analogue of Klyachko's genus to study the equidistribution of $\ell$ and comaj on $[n]\choose k$, generalizing known results on $S_n$.