Séminaire Lotharingien de Combinatoire, 89B.22 (2023), 12 pp.
Joel Brewster Lewis and Eric Marberg
Combinatorial Formulas for Shifted Dual Stable Grothendieck Polynomials
Abstract.
The K-theoretic Schur P- and Q-functions GPλ and GQλ may be concretely defined
as weight generating functions for semistandard shifted set-valued tableaux.
These symmetric functions are the shifted analogues of stable Grothendieck polynomials, and were introduced by Ikeda and Naruse
for applications in geometry.
Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions
gpλ and gqλ via a Cauchy identity involving GPλ and GQλ.
They conjectured that the dual power series are weight generating functions for certain shifted plane partitions.
We prove this conjecture. We also derive a related generating function formula for the images of gpλ and gqλ
under the ω involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author.
Using these results, we verify a conjecture of Ikeda and Naruse that the GQ-functions are a basis for a ring.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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