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

TL;DR

This paper explores the algebraic structure of operators commuting with divided difference operators, introduces new operators generating the full commutant, and connects these to Klyachko's genus and the comaj statistic, advancing Schubert calculus and permutation statistics.

Contribution

It introduces a second set of operators generating the full commutant of divided difference operators and links algebraic, geometric, and combinatorial aspects through Klyachko's genus and comaj statistic.

Findings

The new operators generate the full commutant of divided difference operators.

Klyachko's genus is re-derived using Leibniz combinations of the new operators.

The q-analogue of Klyachko's genus reveals equidistribution of permutation statistics.

Abstract

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 $S_n$.