Hybrid Grothendieck polynomials

For a skew shape λ/μ, we define the hybrid Grothendieck polynomial Gλ/μ(x;t;w) =∑T∈ SVRPP(λ/μ) xircont(T)tceq (T)wex(T) as a weight generating function over set-valued reverse plane partitions of shape λ/μ. It specializes to {itemize} [(1)] the refined stable Grothendieck polynomial introduced by Chan--Pflueger by setting all tᵢ=0; [(2)] the refined dual stable Grothendieck polynomial introduced by Galashin--Grinberg--Liu by setting all wᵢ=0. {itemize} We show that Gλ/μ(x;t;w) is symmetric in the x variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of Gλ/μ(x;t;w) in the basis of Schur functions, extending previous work by Monical--Pechenik--Scrimshaw and Galashin. Based on the Schur expansion, we deduce that hybrid Grothendieck polynomials of straight shapes have saturated Newton polytopes. Finally, using Fomin--Greene's theory on noncommutative Schur functions, we give a combinatorial formula for the image of Gλ/μ(x;t;w) (in the case tᵢ=α and wᵢ=β) under the omega involution on symmetric functions. The formula unifies the structures of weak set-valued tableaux and valued-set tableaux introduced by Lam--Pylyavskyy. Several problems and conjectures are motivated and discussed.