ALGEBRAIC COMBINATORICS

no. 1
Vertex models for Canonical Grothendieck polynomials and their duals
Gunna, Ajeeth1; Zinn-Justin, Paul1
1 School of Mathematics and Statistics University of Melbourne Parkville Victoria 3010 Australia.
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 109-163.

We study exactly solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.235
Classification: 05E05, 82B23
Keywords: Grothendieck polynomials, Exactly solvable lattice models
Gunna, Ajeeth 1; Zinn-Justin, Paul 1

1 School of Mathematics and Statistics University of Melbourne Parkville Victoria 3010 Australia.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gunna, Ajeeth; Zinn-Justin, Paul. Vertex models for Canonical Grothendieck polynomials and their duals. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 109-163. doi : 10.5802/alco.235. https://alco.centre-mersenne.org/articles/10.5802/alco.235/

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

[2] Buciumas, Valentin; Scrimshaw, Travis Double Grothendieck polynomials and colored lattice models, Int. Math. Res. Not. IMRN (2022) no. 10, pp. 7231-7258 | DOI | MR | Zbl

[3] Fomin, S.; Kirillov, A. Yang–Baxter equation, symmetric functions and Grothendieck polynomials, 1993

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

[5] Gunna, Ajeeth; Zinn-Justin, Paul Modified honeycombs in K-Homology of Grassmannian (work in progress.)

[6] Huang, H.Y.; Wu, F.Y.; Kunz, H.; Kim, D. Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model, Physica A, Volume 228 (1996), pp. 1-32 | DOI | MR

[7] Knutson, A.; Zinn-Justin, P. Schubert puzzles and integrability I: invariant trilinear forms, 2017

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

[9] Lascoux, Alain; Naruse, Hiroshi Finite sum Cauchy identity for dual Grothendieck polynomials, Proc. Japan Acad. Ser. A Math. Sci., Volume 90 (2014) no. 7, pp. 87-91 | DOI | MR | Zbl

[10] Lascoux, Alain; Schützenberger, Marcel-Paul Symmetry and flag manifolds, Invariant Theory (Gherardelli, Francesco, ed.), Springer Berlin Heidelberg, Berlin, Heidelberg (1983), pp. 118-144 | DOI | Zbl

[11] Mangazeev, V. On the Yang–Baxter equation for the six-vertex model, Nuclear Phys. B, Volume 882 (2014), pp. 70-96 | DOI | MR | Zbl

[12] McNamara, Peter J. Factorial Grothendieck polynomials, Electron. J. Combin., Volume 13 (2006) no. 1, p. Research Paper 71, 40 pages | DOI | MR | Zbl

[13] Motegi, Kohei; Sakai, Kazumitsu Vertex models, TASEP and Grothendieck polynomials, J. Phys. A, Volume 46 (2013), Paper no. 355201, 26 pages | DOI | MR | Zbl

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

[15] Wheeler, M.; Zinn-Justin, P. Littlewood–Richardson coefficients for Grothendieck polynomials from integrability, J. Reine Angew. Math., Volume 757 (2019), pp. 159-195 | DOI | MR | Zbl

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

[17] Yeliussizov, Damir Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs, J. Comb. Theory, Ser. A, Volume 161 (2019), pp. 453-485 | DOI | MR | Zbl

[18] Yeliussizov, Damir Enumeration of plane partitions by descents, J. Combin. Theory Ser. A, Volume 178 (2021), Paper no. 105367, 18 pages | DOI | MR | Zbl

[19] Zinn-Justin, P. Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009 http://www.lpthe.jussieu.fr/~pzinn/publi/hdr.pdf (Habilitation thesis)

[20] Zinn-Justin, P. Schur functions and Littlewood–Richardson rule from exactly solvable tiling models, 2012 http://www.lpthe.jussieu.fr/~pzinn/semi/berkeley.pdf (Chern–Simons Research Lectures)

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