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 ${\mathrm{\partial }}_i$. 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 $H^{\ast }(Fl(n))\to H^{\ast }($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 $\mathrm{\ell }$ and comaj on $\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$, generalizing known results on $S_n$.