Abstract
Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi–Trudi-type identities, and associated Fomin–Greene operators.
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Notes
We will usually write F or F(x), meaning that it is \(F(x_1, x_2, \ldots )\). Similarly, \(F(x/(1-\alpha x))\) refers to the function \(F(x_1/(1 - \alpha x_1), x_2/(1 - \alpha x_2), \ldots )\). If the function F(x) is of a single variable x, it will be stated or clear from the context.
There is another way of defining \(s_{\lambda }(u)\), directly converting SSYT into monomials consisting of the u variables.
The first relations can be changed to the non-local Knuth relations: \(u_i u_k u_j = u_k u_i u_j, i \le j < k, |i - k| \ge 2\) and \(u_j u_i u_k = u_j u_k u_i, i < j \le k, |i - k| \ge 2\) (which we do not use for our purposes).
These forests have special structure, so not all lattice forests will correspond to the objects that we define here.
The function \(g^{(\alpha , \beta )}_{\lambda /\mu }(x)\) given by that explicit formula coincides exactly with combinatorial definition for \(g^{(\alpha , \beta )}_{\lambda }\) extended for a skew shape.
One can take a more familiar specialization \((\alpha , \beta ) = (q, q^{-1})\).
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Acknowledgments
I am grateful to Pavlo Pylyavskyy for stimulation of this project, many helpful discussions and insightful suggestions. Initial stages of this work began while I was visiting the IMA (Institute for Mathematics and its Applications), and the major part was completed during my visit in the Department of Mathematics at MIT. I am thankful to Richard Stanley and Alexander Postnikov for their hospitality at MIT and for helpful conversations. I am grateful to Askar Dzhumadil’daev for his support and helpful discussions, parts of this work were reported in his seminar. I also thank Sergey Fomin and Victor Reiner for their comments and the referees for helpful suggestions.
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Yeliussizov, D. Duality and deformations of stable Grothendieck polynomials. J Algebr Comb 45, 295–344 (2017). https://doi.org/10.1007/s10801-016-0708-4
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DOI: https://doi.org/10.1007/s10801-016-0708-4