Séminaire Lotharingien de Combinatoire, 91B.96 (2024), 12 pp.

Christian Gaetz, Rebecca Goldin and Allen Knutson

The Commutant of Divided Difference Operators, Klyachko's Genus, and the comaj Statistic

Abstract. Hamaker, Nenashev, Pechenik, Speyer and Weigandt (in various combinations) 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 i that generate the full commutant, and show how a Hopf-algebraic approach naturally reproduces the operators ξν of Nenashev. We then pause to study Klyachko's homomorphism H*(Fl(n)) -> 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 i. We use Nadeau and Tewari's q-analogue of Klyachko's genus to study the equidistribution of ℓ and comaj on ([n]) k, generalizing known results on Sn.

Received: November 15, 2023. Accepted: February 15, 2024. Final version: April 1, 2024.

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