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Duality and deformations of stable Grothendieck polynomials.(English)Zbl 1355.05263

Summary: 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.

MSC:

05E05 Symmetric functions and generalizations
14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds

Cite

References:

[1]Blasiak, J., Fomin, S.: Noncommutative Schur functions, switchboards, and positivity. Preprint arXiv:1510.00657 (2015) ·Zbl 1355.05249
[2]Buch, A.: A Littlewood Richardson rule for the K-theory of Grassmannians. Acta Math. 189, 37-78 (2002) ·Zbl 1090.14015 ·doi:10.1007/BF02392644
[3]Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Theory Ser. A 72, 277-292 (1995) ·Zbl 0839.05093 ·doi:10.1016/0097-3165(95)90065-9
[4]Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discrete Math. 193, 179-200 (1998) ·Zbl 1011.05062 ·doi:10.1016/S0012-365X(98)00140-X
[5]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) ·Zbl 1278.05240
[6]Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153, 123-143 (1996) ·Zbl 0852.05078 ·doi:10.1016/0012-365X(95)00132-G
[7]Galashin, P., Grinberg, D., Liu, G.: Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions. Preprint arXiv:1509.03803 (2015) ·Zbl 1344.05148
[8]Gessel, I., Viennot, X.: Determinants, paths, and plane partitions. Preprint (1989) ·Zbl 0579.05004
[9]Ikeda, T., Naruse, \[H.: KK\]-theoretic analogues of factorial Schur P- and Q-functions. Adv. Math. 243, 22-66 (2013) ·Zbl 1278.05240 ·doi:10.1016/j.aim.2013.04.014
[10]Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. math. 53, 165-184 (1979) ·Zbl 0499.20035 ·doi:10.1007/BF01390031
[11]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)
[12]Lam, T., Pylyavskyy, P.: Combinatorial Hopf algebras and K-homology of Grassmanians. Int. Math. Res. Not. (2007). doi:10.1093/imrn/rnm125 ·Zbl 1134.16017
[13]Lascoux, A., Naruse, H.: Finite sum Cauchy identity for dual Grothendieck polynomials. Proc. Jpn. Acad. Ser. A 90, 87-91 (2014) ·Zbl 1360.05183
[14]Lascoux, A., Schutzenberger, M.-P.: Symmetry and flag manifolds. Lect. Notes Math. 996, 118-144 (1983) ·Zbl 0542.14031 ·doi:10.1007/BFb0063238
[15]Lenart, C.: Combinatorial aspects of the K-theory of Grassmannians. Ann. Comb. 4(1), 67-82 (2000) ·Zbl 0958.05128 ·doi:10.1007/PL00001276
[16]Maconald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1998) ·Zbl 0899.05068
[17]Molev, A.: Comultiplication rules for the double Schur functions and Cauchy identities. Electron. J. Comb. 16, R13 (2009) ·Zbl 1182.05128
[18]Motegi, K., Sakai, K.: Vertex models, TASEP and Grothendieck polynomials. J. Phys. A 46(35), 26 (2013) ·Zbl 1278.82042 ·doi:10.1088/1751-8113/46/35/355201
[19]Patrias, R., Pylyavskyy, P.: K-theoretic Poirier-Reutenauer bialgebra. Preprint arXiv:1404.4340 (2014) ·Zbl 1328.05193
[20]Patrias, R.: Antipode formulas for combinatorial Hopf algebras. Preprint arXiv:1501.00710 (2015) ·Zbl 1351.05234
[21]Shimozono, M., Zabrocki, M.: Stable Grothendieck symmetric functions and \[\Omega\] Ω-calculus. Preprint (2003) ·Zbl 1122.17018
[22]Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999) ·Zbl 0928.05001 ·doi:10.1017/CBO9780511609589
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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