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Set-valued skyline fillings. (English. French summary) Zbl 1384.05160
Summary: Set-valued tableaux play an important role in combinatorial \(K\)-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and we then give analogues of results of J. Haglund et al. [J. Comb. Theory, Ser. A 118, No. 2, 463–490 (2011; Zbl 1229.05270)]. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart’s Schur expansion of the Grothendieck polynomial \(G_\lambda\), using the uncrowding operator of V. Reiner, B. Tenner and A. Yong [“Poset edge densities, nearly reduced words, and barely set-valued tableaux”, Preprint, arXiv:1603.09589].

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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[1] A. Buch. “A Littlewood-Richardson rule for the K-theory of Grassmannians”. Acta Math. 189(2002), pp. 37-78.DOI. · Zbl 1090.14015
[2] M. Demazure. “Désingularisation des variétés de Schubert”. Ann. É. N. S. 4.7 (1974), pp. 53- 88. · Zbl 0312.14009
[3] J. Haglund, M. Haiman, and N. Loehr. “A combinatorial formula for nonsymmetric Mac donald polynomials”. Amer. J. Math. 103.2 (2008), pp. 359-383.DOI. · Zbl 1246.05162
[4] J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. “Quasisymmetric Schur func tions”. J. Combin. Theory Ser. A 118 (2011), pp. 463-490.DOI. · Zbl 1229.05270
[5] J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. “Refinements of the Littlewood Ricardson Rule”. Trans. Amer. Math. Soc. 363 (2011), pp. 1665-1686.DOI. · Zbl 1229.05269
[6] A. Knutson, E. Miller, and A. Yong. “Tableau complexes”. Isr. J. Math. 163.1 (2008), pp. 317- 343.DOI. 12 Cara Monical · Zbl 1145.05055
[7] T. Lam and P. Pylyavskyy. “Combinatorial Hopf Algebras and K-Homology of Grassma nians”. Int. Math. Res. Not. (2007), Art. rnm125.DOI. · Zbl 1134.16017
[8] A. Lascoux. “Transition on Grothendieck polynomials”. Physics and Combinatorics (Nagoya, 2000). World Scientific, 2001, pp. 164-179. · Zbl 1052.14059
[9] A. Lascoux and M.-P. Schützenberger. “Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux”. C.R. Acad. Sci. Paris 295 (1982), pp. 629-633. · Zbl 0542.14030
[10] A. Lascoux and M.-P. Schützenberger. “Keys & standard bases”. Invariant Theory and Tableaux (Minneapolis, MN, 1988). IMA Vol. Math. Appl., Vol. 19. Springer, 1990, pp. 125-144.
[11] C. Lenart. “Combinatorial Aspects of the K-Theory of Grassmannians”. Ann. Comb. 4 (2000), pp. 67-82.DOI. · Zbl 0958.05128
[12] K. Luoto, S. Mykytiuk, and S. van Willigenburg. An Introduction to Quasisymmetric Schur Functions. Springer, 2013. · Zbl 1277.16027
[13] S. Mason. “A decomposition of Schur functions and an analogue of the Robinson-Schensted Knuth algorithm”. Sém. Lothar. Combin. 57 (2008), Art. B57e.URL. · Zbl 1193.05160
[14] S. Mason. “An explicit construction of type A Demazure atoms”. J. Algebraic Combin. 29 (2009), pp. 295-313.DOI. · Zbl 1210.05175
[15] O. Pechenik and A. Yong. “Genomic tableaux”. J. Algebraic Combin. 45 (2016), pp. 649-685. DOI. · Zbl 1362.05134
[16] V. Reiner and M. Shimozono. “Key polynomials and a flagged Littlewood-Richardson rule”. J. Combin. Theory Ser. A 70 (1995), pp. 107-143.DOI. · Zbl 0819.05058
[17] V. Reiner, B. Tenner, and A. Yong. “Poset edge densities, nearly reduced words, and barely set-valued tableaux”. 2016. arXiv:1603.09589. · Zbl 1391.05269
[18] C. Ross and A. Yong. “Combinatorial Rules for Three Bases of Polynomials”. Sém. Lothar. Combin. 74 (2015), Art. B74a.URL. · Zbl 1328.05200
[19] H. Thomas and A. Yong. “A jeu de taquin theory for increasing tableau, with applications to K-theoretic Schubert calculus”. Algebra Number Theory 3 (2009), pp. 121-148.DOI. · Zbl 1229.05285
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