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 \((\alpha , \beta ) = (q, q^{-1})\).
References
Blasiak, J., Fomin, S.: Noncommutative Schur functions, switchboards, and positivity. Preprint arXiv:1510.00657 (2015)
Buch, A.: A Littlewood Richardson rule for the K-theory of Grassmannians. Acta Math. 189, 37–78 (2002)
Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Theory Ser. A 72, 277–292 (1995)
Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discrete Math. 193, 179–200 (1998)
Fomin, S., Kirillov, A. N.: Grothendieck polynomials and the Yang–Baxter equation. In: Proceedings of 6th International Conference on Formal Power Series and Algebraic Combinatorics, DIMACS, pp. 183–190 (1994)
Fomin, S., Kirillov, A.N.: The Yang–Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153, 123–143 (1996)
Galashin, P., Grinberg, D., Liu, G.: Refined dual stable Grothendieck polynomials and generalized Bender–Knuth involutions. Preprint arXiv:1509.03803 (2015)
Gessel, I., Viennot, X.: Determinants, paths, and plane partitions. Preprint (1989)
Ikeda, T., Naruse, H.: \(K\)-theoretic analogues of factorial Schur P- and Q-functions. Adv. Math. 243, 22–66 (2013)
Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. math. 53, 165–184 (1979)
Kirillov, A.: On some quadratic algebras I 1/2: combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials. Preprint RIMS-1817 arXiv:1502.00426 (2015)
Lam, T., Pylyavskyy, P.: Combinatorial Hopf algebras and K-homology of Grassmanians. Int. Math. Res. Not. (2007). doi:10.1093/imrn/rnm125
Lascoux, A., Naruse, H.: Finite sum Cauchy identity for dual Grothendieck polynomials. Proc. Jpn. Acad. Ser. A 90, 87–91 (2014)
Lascoux, A., Schutzenberger, M.-P.: Symmetry and flag manifolds. Lect. Notes Math. 996, 118–144 (1983)
Lenart, C.: Combinatorial aspects of the K-theory of Grassmannians. Ann. Comb. 4(1), 67–82 (2000)
Maconald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1998)
Molev, A.: Comultiplication rules for the double Schur functions and Cauchy identities. Electron. J. Comb. 16, R13 (2009)
Motegi, K., Sakai, K.: Vertex models, TASEP and Grothendieck polynomials. J. Phys. A 46(35), 26 (2013)
Patrias, R., Pylyavskyy, P.: K-theoretic Poirier–Reutenauer bialgebra. Preprint arXiv:1404.4340 (2014)
Patrias, R.: Antipode formulas for combinatorial Hopf algebras. Preprint arXiv:1501.00710 (2015)
Shimozono, M., Zabrocki, M.: Stable Grothendieck symmetric functions and \(\Omega \)-calculus. Preprint (2003)
Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
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