ALGEBRAIC COMBINATORICS

no. 3
K-theoretic crystals for set-valued tableaux of rectangular shapes
Pechenik, Oliver1; Scrimshaw, Travis2
1 Department of Combinatorics & Optimization University of Waterloo Waterloo ON N2L 3G1, Canada
2 School of Mathematics and Physics The University of Queensland St. Lucia QLD 4072, Australia
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536.

In earlier work with C. Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials L wλ when λ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross–Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.221
Classification: 05E05, 05A19, 14M15, 17B37
Keywords: Grothendieck polynomial, crystal, Lascoux polynomial, quantum group, set-valued tableau, Kohnert move, skyline tableau.
Pechenik, Oliver 1; Scrimshaw, Travis 2

1 Department of Combinatorics & Optimization University of Waterloo Waterloo ON N2L 3G1, Canada
2 School of Mathematics and Physics The University of Queensland St. Lucia QLD 4072, Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Pechenik, Oliver; Scrimshaw, Travis. K-theoretic crystals for set-valued tableaux of rectangular shapes. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536. doi : 10.5802/alco.221. https://alco.centre-mersenne.org/articles/10.5802/alco.221/

[1] Andersen, Henning H. Schubert varieties and Demazure’s character formula, Invent. Math., Volume 79 (1985) no. 3, pp. 611-618 | DOI | MR | Zbl

[2] Assaf, Sami; Searles, Dominic Kohnert tableaux and a lifting of quasi-Schur functions, J. Combin. Theory Ser. A, Volume 156 (2018), pp. 85-118 | DOI | MR | Zbl

[3] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | DOI | MR | Zbl

[4] Bressler, Paul; Evens, Sam The Schubert calculus, braid relations, and generalized cohomology, Trans. Amer. Math. Soc., Volume 317 (1990) no. 2, pp. 799-811 | DOI | MR | Zbl

[5] Buch, Anders S. A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | DOI | MR | Zbl

[6] Buciumas, Valentin; Scrimshaw, Travis; Weber, Katherine Colored five-vertex models and Lascoux polynomials and atoms, J. Lond. Math. Soc. (2), Volume 102 (2020) no. 3, pp. 1047-1066 | DOI | MR | Zbl

[7] Bump, Daniel; Schilling, Anne Crystal bases. Representations and Combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, xii+279 pages | DOI | MR | Zbl

[8] Davis, Michael W. The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, 32, Princeton University Press, Princeton, NJ, 2008, xvi+584 pages | MR | Zbl

[9] Demazure, Michel Une nouvelle formule des caractères, Bull. Sci. Math. (2), Volume 98 (1974) no. 3, pp. 163-172 | MR | Zbl

[10] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang–Baxter equation, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, 1994, pp. 183-189 | MR

[11] Fulton, William Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR | Zbl

[12] Ganter, Nora; Ram, Arun Generalized Schubert calculus, J. Ramanujan Math. Soc., Volume 28A (2013), pp. 149-190 | MR | Zbl

[13] Graham, William; Kreiman, Victor Excited Young diagrams, equivariant K-theory, and Schubert varieties, Trans. Amer. Math. Soc., Volume 367 (2015) no. 9, pp. 6597-6645 | DOI | MR | Zbl

[14] Hudson, Thomas A Thom–Porteous formula for connective K-theory using algebraic cobordism, J. K-Theory, Volume 14 (2014) no. 2, pp. 343-369 | DOI | MR | Zbl

[15] Hudson, Thomas; Ikeda, Takeshi; Matsumura, Tomoo; Naruse, Hiroshi Degeneracy loci classes in K-theory—determinantal and Pfaffian formula, Adv. Math., Volume 320 (2017), pp. 115-156 | DOI | MR | Zbl

[16] Hudson, Thomas; Matsumura, Tomoo Vexillary degeneracy loci classes in K-theory and algebraic cobordism, European J. Combin., Volume 70 (2018), pp. 190-201 | DOI | MR | Zbl

[17] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | DOI | MR | Zbl

[18] Kane, Richard Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5, Springer-Verlag, New York, 2001, x+379 pages | DOI | MR | Zbl

[19] Kashiwara, Masaki Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys., Volume 133 (1990) no. 2, pp. 249-260 | DOI | MR | Zbl

[20] Kashiwara, Masaki On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR | Zbl

[21] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | DOI | MR | Zbl

[22] Kashiwara, Masaki; Nakashima, Toshiki Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, Volume 165 (1994) no. 2, pp. 295-345 | DOI | MR | Zbl

[23] Kirillov, Anatol N. Notes on Schubert, Grothendieck and key polynomials, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 12 (2016), Paper no. 034, 56 pages | DOI | MR | Zbl

[24] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Volume 630 (2009), pp. 1-31 | DOI | MR | Zbl

[25] Kohnert, Axel Weintrauben, Polynome, Tableaux, Bayreuth. Math. Schr. (1991) no. 38, pp. 1-97 (Dissertation, Universität Bayreuth, Bayreuth, 1990) | MR | Zbl

[26] Lakshmibai, Venkatramani; Musili, C.; Seshadri, Conjeeveram S. Geometry of G/P. IV. Standard monomial theory for classical types, Proc. Indian Acad. Sci. Sect. A Math. Sci., Volume 88 (1979) no. 4, pp. 279-362 | DOI | MR | Zbl

[27] Lascoux, Alain Anneau de Grothendieck de la variété de drapeaux, The Grothendieck Festschrift, Vol. III (Progr. Math.), Volume 88, Birkhäuser Boston, Boston, MA, 1990, pp. 1-34 | DOI | MR | Zbl

[28] Lascoux, Alain Transition on Grothendieck polynomials, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 164-179 | DOI | MR | Zbl

[29] Lascoux, Alain; Schützenberger, Marcel-Paul Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 | MR | Zbl

[30] Lascoux, Alain; Schützenberger, Marcel-Paul Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) (Lecture Notes in Math.), Volume 996, Springer, Berlin, 1983, pp. 118-144 | DOI | MR | Zbl

[31] Lascoux, Alain; Schützenberger, Marcel-Paul Keys & standard bases, Invariant theory and tableaux (Minneapolis, MN, 1988) (IMA Vol. Math. Appl.), Volume 19, Springer, New York, 1990, pp. 125-144 | MR | Zbl

[32] Lenart, Cristian On the combinatorics of crystal graphs. I. Lusztig’s involution, Adv. Math., Volume 211 (2007) no. 1, pp. 204-243 | DOI | MR | Zbl

[33] Lenart, Cristian; Zainoulline, Kirill Towards generalized cohmology Schubert calculus via formal root polynomials, Math. Res. Lett., Volume 24 (2017) no. 3, pp. 839-877 | DOI | MR | Zbl

[34] Littelmann, Peter Crystal graphs and Young tableaux, J. Algebra, Volume 175 (1995) no. 1, pp. 65-87 | DOI | MR | Zbl

[35] Mason, Sarah A decomposition of Schur functions and an analogue of the Robinson–Schensted–Knuth algorithm, Sém. Lothar. Combin., Volume 57 (2006/08), Paper no. B57e, 24 pages | MR | Zbl

[36] Matsumura, Tomoo Flagged Grothendieck polynomials, J. Algebraic Combin., Volume 49 (2019) no. 3, pp. 209-228 | DOI | MR | Zbl

[37] Matsumura, Tomoo; Sugimoto, Shogo Factorial Flagged Grothendieck Polynomials, Schubert Calculus and Its Applications in Combinatorics and Representation Theory, ICTSC 2017 (Hu, J.; Li, C.; Mihalcea, L. C., eds.) (Proceedings in Mathematics and Statistics), Volume 332, Springer, Singapore, 2020, pp. 1-15 | DOI | MR | Zbl

[38] Monical, Cara Set-valued skyline fillings (2016) (https://arxiv.org/abs/1611.08777)

[39] Monical, Cara; Pechenik, Oliver; Scrimshaw, Travis Crystal structures for symmetric Grothendieck polynomials, Transform. Groups, Volume 26 (2021) no. 3, pp. 1025-1075 | DOI | MR | Zbl

[40] Monical, Cara; Pechenik, Oliver; Searles, Dominic Polynomials from combinatorial K-theory, Canad. J. Math., Volume 73 (2021) no. 1, pp. 29-62 | DOI | MR | Zbl

[41] Postnikov, Alexander; Stanley, Richard P. Chains in the Bruhat order, J. Algebraic Combin., Volume 29 (2009) no. 2, pp. 133-174 | DOI | MR | Zbl

[42] Proctor, Robert A.; Willis, Matthew J. Semistandard tableaux for Demazure characters (key polynomials) and their atoms, European J. Combin., Volume 43 (2015), pp. 172-184 | DOI | MR | Zbl

[43] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood–Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | DOI | MR | Zbl

[44] Ross, Colleen Combinatorial formulae for Grothendieck-Demazure and Grothendieck polynomials (2011) (http://www.math.uiuc.edu/~ayong/student_projects/Ross.pdf) (Technical report)

[45] Ross, Colleen; Yong, Alexander Combinatorial rules for three bases of polynomials, Sém. Lothar. Combin., Volume 74 (2015), Paper no. B74a, 11 pages | MR | Zbl

[46] Ross, Colleen; Yong, Alexander Combinatorial rules for three bases of polynomials (2017) https://faculty.math.illinois.edu/~ayong/polynomials.Seminaire.revision.2017.pdf (Unpublished erratum to [45])

[47] community, The Sage-Combinat Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008 (http://combinat.sagemath.org)

[48] Sage Mathematics Software (Version 8.6) (2019) http://www.sagemath.org

[49] Stanley, Richard P. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl

[50] Stembridge, John R. On the fully commutative elements of Coxeter groups, J. Algebraic Combin., Volume 5 (1996) no. 4, pp. 353-385 | DOI | MR | Zbl

[51] Willis, Matthew J. A direct way to find the right key of a semistandard Young tableau, Ann. Comb., Volume 17 (2013) no. 2, pp. 393-400 | DOI | MR | Zbl

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