Abstract
The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.
Highlights
Thomas Lam and Pavlo Pylyavskyy, in [LamPyl07, §9.1], studied dual stable Grothendieck polynomials, a deformation of the Schur functions
We devise a common generalization of the dual stable Grothendieck polynomial gλ/μ and the classical skew Schur function sλ/μ
In addition, the values of T increase strictly down each column, T is called a semistandard tableau of shape λ/μ. (See Fulton’s [Fulton97] for an exposition of properties and applications of semistandard tableaux.) We denote the set of all reverse plane partitions of shape λ/μ by RPP (λ/μ)
Summary
Thomas Lam and Pavlo Pylyavskyy, in [LamPyl07, §9.1], (and earlier Mark Shimozono and Mike Zabrocki in unpublished work of 2003) studied dual stable Grothendieck polynomials, a deformation (in a sense) of the Schur functions. The dual stable Grothendieck polynomial gλ/μ is a generating function for the reverse plane partitions of shape λ/μ; these, unlike semistandard tableaux, are only required to have their entries increase weakly down columns (and along rows). As proven in [LamPyl07, §9.1], this power series gλ/μ is a symmetric function (albeit, unlike sλ/μ, an inhomogeneous one in general). We devise a common generalization of the dual stable Grothendieck polynomial gλ/μ and the classical skew Schur function sλ/μ.
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