ALGEBRAIC COMBINATORICS

no. 1
Free fermions and canonical Grothendieck polynomials
Iwao, Shinsuke1; Motegi, Kohei2; Scrimshaw, Travis3
1 Faculty of Business and Commerce Keio University Hiyosi 4–1–1, Kohoku-ku, Yokohama-si Kanagawa 223-8521 (Japan)
2 Faculty of Marine Technology Tokyo University of Marine Science and Technology Etchujima 2–1–6, Koto-Ku Tokyo, 135-8533 (Japan)
3 Faculty of Science Hokkaido University 5 Chōme Kita 8 Jōnishi, Kita Ward Sapporo, Hokkaidō 060-0808 (Japan)
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 245-274.

We give a presentation of refined (dual) canonical Grothendieck polynomials and their skew versions using free fermions. Using this, we derive a number of identities, including the skew Cauchy identities, branching rules, expansion formulas, and integral formulas.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.332
Classification: 05E05, 82B23, 14M15, 05A19
Keywords: Grothendieck polynomial, free fermion
Iwao, Shinsuke 1; Motegi, Kohei 2; Scrimshaw, Travis 3

1 Faculty of Business and Commerce Keio University Hiyosi 4–1–1, Kohoku-ku, Yokohama-si Kanagawa 223-8521 (Japan)
2 Faculty of Marine Technology Tokyo University of Marine Science and Technology Etchujima 2–1–6, Koto-Ku Tokyo, 135-8533 (Japan)
3 Faculty of Science Hokkaido University 5 Chōme Kita 8 Jōnishi, Kita Ward Sapporo, Hokkaidō 060-0808 (Japan)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Iwao, Shinsuke; Motegi, Kohei; Scrimshaw, Travis. Free fermions and canonical Grothendieck polynomials. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 245-274. doi : 10.5802/alco.332. https://alco.centre-mersenne.org/articles/10.5802/alco.332/

[1] Alexandrov, Alexander; Zabrodin, Anton Free fermions and tau-functions, J. Geom. Phys., Volume 67 (2013), pp. 37-80 | DOI | MR | Zbl

[2] Amanov, Alimzhan; Yeliussizov, Damir Determinantal formulas for dual Grothendieck polynomials, Proc. Amer. Math. Soc., Volume 150 (2022) no. 10, pp. 4113-4128 | DOI | MR | Zbl

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

[4] Chan, Melody; Pflueger, Nathan Combinatorial relations on skew Schur and skew stable Grothendieck polynomials, Algebr. Comb., Volume 4 (2021) no. 1, pp. 175-188 | DOI | Numdam | MR | Zbl

[5] Date, Etsurō; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji Transformation groups for soliton equations, Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, pp. 39-119 | MR | Zbl

[6] Dieker, A. B.; Warren, J. Determinantal transition kernels for some interacting particles on the line, Ann. Inst. Henri Poincaré Probab. Stat., Volume 44 (2008) no. 6, pp. 1162-1172 | DOI | Numdam | MR | Zbl

[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

[8] Galashin, Pavel; Grinberg, Darij; Liu, Gaku Refined dual stable Grothendieck polynomials and generalized Bender–Knuth involutions, Electron. J. Combin., Volume 23 (2016) no. 3, Paper no. 3.14, 28 pages | MR | Zbl

[9] Gunna, Ajeeth; Zinn-Justin, Paul Vertex models for Canonical Grothendieck polynomials and their duals, Algebr. Comb., Volume 6 (2023) no. 1, pp. 109-162 | DOI | MR | Zbl

[10] Hardt, Andrew Lattice models, Hamiltonian operators, and symmetric functions, 2021 | arXiv

[11] 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

[12] Hwang, Byung-Hak; Jang, Jihyeug; Kim, Jang Soo; Song, Minho; Song, U-Keun Refined canonical stable Grothendieck polynomials and their duals, 2021 | arXiv

[13] Iwao, Shinsuke Grothendieck polynomials and the boson-fermion correspondence, Algebr. Comb., Volume 3 (2020) no. 5, pp. 1023-1040 | DOI | Numdam | MR | Zbl

[14] Iwao, Shinsuke Free-fermions and skew stable Grothendieck polynomials, J. Algebraic Combin., Volume 56 (2022) no. 2, pp. 493-526 | DOI | MR | Zbl

[15] Iwao, Shinsuke Free fermions and Schur expansions of multi-Schur functions, J. Combin. Theory Ser. A, Volume 198 (2023), Paper no. 105767, 23 pages | DOI | MR | Zbl

[16] Iwao, Shinsuke; Motegi, Kohei; Scrimshaw, Travis Combinatorial description of canonical free fermions (2024) (In preparation)

[17] Iwao, Shinsuke; Motegi, Kohei; Scrimshaw, Travis Free fermionic probability theory and K-theoretic Schubert calculus (2024) (In preparation)

[18] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990, xxii+400 pages | DOI | MR

[19] Kac, Victor G.; Raina, Ashok K.; Rozhkovskaya, Natasha Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Advanced Series in Mathematical Physics, 29, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013, xii+237 pages | DOI | MR

[20] Kim, Jang Soo Jacobi–Trudi formula for refined dual stable Grothendieck polynomials, J. Combin. Theory Ser. A, Volume 180 (2021), Paper no. 105415, 33 pages | DOI | MR | Zbl

[21] Kim, Jang Soo Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials, Algebr. Comb., Volume 5 (2022) no. 1, pp. 121-148 | DOI | Numdam | MR | Zbl

[22] Lascoux, Alain Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, 99, Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003, xii+268 pages | DOI | MR

[23] 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

[24] 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

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

[26] Lenart, Cristian; Postnikov, Alexander A combinatorial model for crystals of Kac–Moody algebras, Trans. Amer. Math. Soc., Volume 360 (2008) no. 8, pp. 4349-4381 | DOI | MR | Zbl

[27] Loehr, Nicholas A.; Remmel, Jeffrey B. A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | DOI | MR | Zbl

[28] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR

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

[30] Miwa, T.; Jimbo, M.; Date, E. Solitons: Differential equations, symmetries and infinite-dimensional algebras, Cambridge Tracts in Mathematics, 135, Cambridge University Press, Cambridge, 2000, x+108 pages | MR

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

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

[33] Motegi, Kohei; Scrimshaw, Travis Refined dual Grothendieck polynomials, integrability, and the Schur measure, 2020 | arXiv

[34] Pan, Jianping; Pappe, Joseph; Poh, Wencin; Schilling, Anne Uncrowding algorithm for hook-valued tableaux, Ann. Comb., Volume 26 (2022) no. 1, pp. 261-301 | DOI | MR | Zbl

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

[36] Sage Mathematics Software (Version 9.7) (2022) https://www.sagemath.org

[37] Warnaar, S. Ole Remarks on the paper “Skew Pieri rules for Hall–Littlewood functions” by Konvalinka and Lauve, J. Algebraic Combin., Volume 38 (2013) no. 3, pp. 519-526 | DOI | MR | Zbl

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

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

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

[41] Yeliussizov, Damir Dual Grothendieck polynomials via last-passage percolation, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 4, pp. 497-503 | Numdam | MR | Zbl

[42] Zinn-Justin, Paul Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009 (Habilitation thesis)

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