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no. 5
Grothendieck polynomials and the boson-fermion correspondence
Iwao, Shinsuke1
1 Department of Mathematics, Tokai University, 4-1-1, Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan.
Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1023-1040.

In this paper we study algebraic and combinatorial properties of symmetric Grothendieck polynomials and their dual polynomials by means of the boson-fermion correspondence. We show that these symmetric functions can be expressed as a vacuum expectation value of some operator that is written in terms of free-fermions. By using the free-fermionic expressions, we obtain alternative proofs of determinantal formulas and Pieri type formulas.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.116
Classification: 05E05, 05E10, 17B69
Keywords: Symmetric Grothendieck polynomials, Boson-fermion correspondence.

Iwao, Shinsuke 1

1 Department of Mathematics, Tokai University, 4-1-1, Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Iwao, Shinsuke. Grothendieck polynomials and the boson-fermion correspondence. Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1023-1040. doi : 10.5802/alco.116. https://alco.centre-mersenne.org/articles/10.5802/alco.116/

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

[2] 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

[3] Fomin, Sergey; Greene, Curtis Noncommutative Schur functions and their applications, Discrete Math., Volume 193 (1998) no. 1-3, pp. 179-200 | DOI | MR | Zbl

[4] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang–Baxter equation, Proc. Formal Power Series and Alg. Comb (1994), pp. 183-190

[5] Fulton, William Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, Cambridge University Press, 1996 | DOI | Zbl

[6] 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

[7] Ikeda, Takeshi; Naruse, Hiroshi K-theoretic analogues of factorial Schur P- and Q-functions, Adv. Math., Volume 243 (2013), pp. 22-66 | DOI | MR | Zbl

[8] Iwao, Shinsuke; Nagai, Hidetomo The discrete Toda equation revisited: dual β-Grothendieck polynomials, ultradiscretization, and static solitons, J. Phys. A, Volume 51 (2018) no. 13, Paper no. 134002 | DOI | MR | Zbl

[9] Jimbo, Michio; Miwa, Tetsuji Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., Volume 19 (1983) no. 3, pp. 943-1001 | DOI | MR | Zbl

[10] 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, 2013 | MR | Zbl

[11] Kirillov, Anatol N. On some quadratic algebras I 1 2: combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss–Catalan, universal Tutte and reduced polynomials, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 12 (2016), Paper no. Paper no. 002, 172 pages | DOI | MR | Zbl

[12] 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

[13] 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

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

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

[16] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford university press, 1998 | Zbl

[17] Miwa, Tetsuji; Jimbo, Michio; Date, Etsuro; Reid, Miles Solitons: Differential equations, symmetries and infinite dimensional algebras, Cambridge Tracts in Mathematics, 135, Cambridge University Press, 2012

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

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

[20] Nakagawa, Masaki; Naruse, Hiroshi Universal factorial Schur P,Q-functions and their duals (2018) (https://arxiv.org/abs/1812.03328)

[21] Shimozono, Mark; Zabrocki, Mike Stable Grothendieck polynomials and Ω-calculus (2011) (unpublished)

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

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