Séminaire Lotharingien de Combinatoire, 89B.66 (2023), 12 pp.

Eric Marberg and Travis Scrimshaw

Shifted key polynomials

Abstract. We introduce shifted analogues of key (resp. atom) polynomials that we call P- and Q-key (resp. atom) polynomials. These families are defined in terms of isobaric divided difference operators applied to dominant symplectic and orthogonal Schubert polynomials. We establish a number of fundamental properties of these functions, formally similar to classical results on key polynomials. For example, we show that our shifted key polynomials are partial versions of Schur P- and Q-functions in a precise sense. We conjecture that symplectic/orthogonal Schubert polynomials expand positively in terms of P/Q-key polynomials. As evidence for this conjecture, we also show that shifted key polynomials are the characters of certain shifted analogues of Demazure crystals.


Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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