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, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, 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
Grothendieck polynomials were introduced by Lascoux and Schutzenberger [LasSch82] to model classes of structure sheaves of Schubert varieties in the K-theory of flag manifolds
Stable Grothendieck polynomials were introduced by Fomin and Kirillov [FomKir96] as the electronic journal of combinatorics 23(3) (2016), #P3.14 stable limits of certain sequences of Grothendieck polynomials
Lam and Pylyavskyy related these functions to the K-homology of the Grassmannian and gave a combinatorial definition of these functions in terms of reverse plane partitions
Summary
Grothendieck polynomials were introduced by Lascoux and Schutzenberger [LasSch82] to model classes of structure sheaves of Schubert varieties in the K-theory of flag manifolds. The second proof reflects back on the first, in particular providing an alternative definition of the generalized Bender-Knuth involutions constructed in the first proof, and showing that these involutions are “the only reasonable choice” in a sense that we clarify We notice that both our the electronic journal of combinatorics 23(3) (2016), #P3.14 proofs are explicitly bijective, unlike the proof of [LamPyl, Theorem 9.1] which relies on computations in an algebra of operators. An extended abstract of this paper, omitting the proofs, is to appear as [GaGrLi16]
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