Crystal structures for canonical Grothendieck functions
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 727-755.

We give a U q (𝔰𝔩 n )-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions.

Received: 2019-09-20
Revised: 2020-02-03
Accepted: 2020-02-06
Published online: 2020-06-02
DOI: https://doi.org/10.5802/alco.111
Classification: 05E05,  05A19,  14M15,  17B37
Keywords: Canonical Grothendieck function, crystal, quantum group, multiset-valued tableau, hook-valued tableau, valued-set tableau.
@article{ALCO_2020__3_3_727_0,
     author = {Hawkes, Graham and Scrimshaw, Travis},
     title = {Crystal structures for canonical Grothendieck functions},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {3},
     year = {2020},
     pages = {727-755},
     doi = {10.5802/alco.111},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_3_727_0/}
}
Hawkes, Graham; Scrimshaw, Travis. Crystal structures for canonical Grothendieck functions. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 727-755. doi : 10.5802/alco.111. https://alco.centre-mersenne.org/item/ALCO_2020__3_3_727_0/

[1] Bandlow, Jason; Morse, Jennifer Combinatorial expansions in K-theoretic bases, Electron. J. Combin., Volume 19 (2012) no. 4, Paper 39, 27 pages | MR 3007174 | Zbl 1267.05037

[2] Benkart, Georgia; Colmenarejo, Laura; Harris, Pamela E.; Orellana, Rosa; Panova, Greta; Schilling, Anne; Yip, Martha A minimaj-preserving crystal on ordered multiset partitions, Adv. in Appl. Math., Volume 95 (2018), pp. 96-115 | Article | MR 3759213 | Zbl 1379.05015

[3] Buch, Anders Skovsted A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | Article | MR 1946917 | Zbl 1090.14015

[4] Buch, Anders Skovsted; Kresch, Andrew; Shimozono, Mark; Tamvakis, Harry; Yong, Alexander Stable Grothendieck polynomials and K-theoretic factor sequences, Math. Ann., Volume 340 (2008) no. 2, pp. 359-382 | Article | MR 2368984 | Zbl 1157.14036

[5] Buch, Anders Skovsted; Samuel, Matthew J. K-theory of minuscule varieties, J. Reine Angew. Math., Volume 719 (2016), pp. 133-171 | Article | MR 3552494 | Zbl 06636676

[6] Bump, Daniel; Schilling, Anne Crystal bases: Representations and Combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, xii+279 pages | Article | MR 3642318 | Zbl 06690908

[7] 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 2307216

[8] Fomin, Sergey; Kirillov, Anatol N. The Yang–Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., Volume 153 (1996) no. 1-3, pp. 123-143 | Article | MR 1394950 | Zbl 0852.05078

[9] Gaetz, Christian; Mastrianni, Michelle; Patrias, Rebecca; Peck, Hailee; Robichaux, Colleen; Schwein, David; Tam, Ka Yu K-Knuth equivalence for increasing tableaux, Electron. J. Combin., Volume 23 (2016) no. 1, Paper 1.40, 37 pages | MR 3484745 | Zbl 1332.05148

[10] Galashin, Pavel A Littlewood–Richardson rule for dual stable Grothendieck polynomials, J. Combin. Theory Ser. A, Volume 151 (2017), pp. 23-35 | Article | MR 3663486 | Zbl 1366.05116

[11] Hudson, Thomas A Thom–Porteous formula for connective K-theory using algebraic cobordism, J. K-Theory, Volume 14 (2014) no. 2, pp. 343-369 | Article | MR 3319705 | Zbl 1314.14015

[12] Ikeda, Takeshi; Iwao, Shinsuke; Maeno, Toshiaki Peterson isomorphism in K-theory and relativistic Toda lattice (2018) (To appear in Int. Math. Res. Not. IMRN) | Article

[13] Ikeda, Takeshi; Naruse, Hiroshi Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009) no. 10, pp. 5193-5221 | Article | MR 2515809 | Zbl 1229.05287

[14] Ikeda, Takeshi; Shimazaki, Tatsushi A proof of K-theoretic Littlewood–Richardson rules by Bender–Knuth-type involutions, Math. Res. Lett., Volume 21 (2014) no. 2, pp. 333-339 | Article | MR 3247060 | Zbl 1301.05358

[15] Iwao, Shinsuke Grothendieck polynomials and the boson-fermion correspondence (2019) (Preprint, https://arxiv.org/abs/1905.07692)

[16] Kashiwara, Masaki Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys., Volume 133 (1990) no. 2, pp. 249-260 | Article | MR 1090425 | Zbl 0724.17009

[17] Kashiwara, Masaki On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | Article | MR 1115118 | Zbl 0749.17017

[18] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | Article | MR 1240605 | Zbl 0794.17008

[19] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not. IMRN, Volume 2007 (2007) no. 24, Art. ID rnm125, 48 pages | Article | MR 2377012 | Zbl 1134.16017

[20] Lascoux, Alain; Leclerc, Bernard; Thibon, Jean-Yves The Plactic Monoid, Algebraic Combinatorics on Words (Lothaire, M., ed.), Cambridge University Press, Cambridge, 2002

[21] 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 686357 | Zbl 0542.14030

[22] Lascoux, Alain; SchĂĽtzenberger, Marcel-Paul Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) (Lecture Notes in Math.) Volume 996, Springer, Berlin, 1983, pp. 118-144 | Article | MR 718129 | Zbl 0542.14031

[23] Lenart, Cristian Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82 | Article | MR 1763950 | Zbl 0958.05128

[24] Lenart, Cristian On the combinatorics of crystal graphs. I. Lusztig’s involution, Adv. Math., Volume 211 (2007) no. 1, pp. 204-243 | Article | MR 2313533 | Zbl 1129.05058

[25] Matsumoto, Hideya Générateurs et relations des groupes de Weyl généralisés, C. R. Acad. Sci. Paris, Volume 258 (1964), pp. 3419-3422 | MR 0183818 | Zbl 0128.25202

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

[27] Monical, Cara; Pechenik, Oliver; Scrimshaw, Travis Crystal structures for symmetric Grothendieck polynomials (2018) (Preprint, http://arxiv.org/abs/1807.03294)

[28] Monical, Cara; Pechenik, Oliver; Searles, Dominic Polynomials from combinatorial K-theory (2019) (To appear in Canad. J. Math., http://arxiv.org/abs/1806.03802) | Article | Zbl 07181206

[29] Motegi, Kohei; Sakai, Kazumitsu Vertex models, TASEP and Grothendieck polynomials, J. Phys. A, Volume 46 (2013) no. 35, 355201, 26 pages | Article | MR 3100873 | Zbl 1278.82042

[30] Motegi, Kohei; Sakai, Kazumitsu K-theoretic boson-fermion correspondence and melting crystals, J. Phys. A, Volume 47 (2014) no. 44, 445202, 30 pages | Article | MR 3270564 | Zbl 1310.82049

[31] Patrias, Rebecca; Pylyavskyy, Pavlo Combinatorics of K-theory via a K-theoretic Poirier–Reutenauer bialgebra, Discrete Math., Volume 339 (2016) no. 3, pp. 1095-1115 | Article | MR 3433916 | Zbl 1328.05193

[32] Pechenik, Oliver; Scrimshaw, Travis K-theoretic crystals for set-valued tableaux of rectangular shape (2019) (Preprint, http://arxiv.org/abs/1904.09674)

[33] Pechenik, Oliver; Searles, Dominic Decompositions of Grothendieck polynomials, Int. Math. Res. Not. IMRN, Volume 2019 (2019) no. 10, pp. 3214-3241 | Article | MR 3952563 | Zbl 07067193

[34] Pechenik, Oliver; Yong, Alexander Genomic tableaux, J. Algebraic Combin., Volume 45 (2017) no. 3, pp. 649-685 | Article | MR 3627499 | Zbl 1362.05134

[35] Pylyavskyy, Pavlo; Yang, Jed Puzzles in K-homology of Grassmannians, Pacific J. Math., Volume 303 (2019) no. 2, pp. 703-727 | Article | MR 4046973 | Zbl 07179021

[36] Reiner, Victor; Tenner, Bridget Eileen; Yong, Alexander Poset edge densities, nearly reduced words, and barely set-valued tableaux, J. Combin. Theory Ser. A, Volume 158 (2018), pp. 66-125 | Article | MR 3800124 | Zbl 1391.05269

[37] Sage-Combinat community Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008 (https://wiki.sagemath.org/combinat) | Article | MR 1676282 | Zbl 0928.05001

[38] Sage Developers Sage Mathematics Software (Version 8.8), 2019 (https://www.sagemath.org) | Article | MR 1997585 | Zbl 1047.17007

[39] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Volume 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | Article | MR 1676282 | Zbl 0928.05001

[40] Stembridge, John R. A local characterization of simply-laced crystals, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, pp. 4807-4823 | Article | MR 1997585 | Zbl 1047.17007

[41] Thomas, Hugh; Yong, Alexander A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus, Algebra Number Theory, Volume 3 (2009) no. 2, pp. 121-148 | Article | MR 2491941 | Zbl 1229.05285

[42] Thomas, Hugh; Yong, Alexander Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm, Adv. in Appl. Math., Volume 46 (2011) no. 1-4, pp. 610-642 | Article | MR 2794040 | Zbl 1227.05262

[43] Wheeler, Michael; Zinn-Justin, Paul Littlewood–Richardson coefficients for Grothendieck polynomials from integrability, J. Reine Angew. Math., Volume 757 (2019), pp. 159-195 | Article | MR 4036573 | Zbl 1428.05323

[44] Yeliussizov, Damir Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., Volume 45 (2017) no. 1, pp. 295-344 | Article | MR 3591379 | Zbl 1355.05263