×

zbMATH — the first resource for mathematics

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].

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
PDF BibTeX XML Cite
Full Text: Link arXiv
References:
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.