Geodesic
Extended plethystic vertex operators and plethystic universal characters
Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 276-290 Cet article a éte moissonné depuis la source Math-Net.Ru

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By means of plethystic-type fermions and plethystic-type boson–fermion correspondence, which is a generalization of the classical boson–fermion correspondence, we obtain a two-component twisted plethystic-type symmetric functions $S_{[\lambda,\mu]}^{(\alpha,\beta)}$ from an $(\alpha,\beta)$-type boson–fermion correspondence, similarly to how the universal character $S_{[\lambda,\mu]}$ is derived from the classical boson–fermion correspondence (the twisted Jacobi–Trudi formula). As a generalization of the universal character hierarchy, we then construct the $(\alpha,\beta)$-type plethystic universal character hierarchy that contains a series of nonlinear partial differential equations of infinite order, and obtain its tau functions and Plücker relations.
Mots-clés : boson–fermion correspondence, plethystic-type fermions, Plücker relation.
Keywords: plethystic-type symmetric functions, plethystic universal character hierarchy 
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Chuanzhong Li; Yong Zhang; Huanhe Dong. Extended plethystic vertex operators and plethystic universal characters. Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 276-290. http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a7/

[1] E. Date, M. Kashiwara, M. Jimbo, T. Miwa, “Transformation groups for soliton equations”, Non-linear Integrable Systems – Classical Theory and Quantum Theory (Kyoto, Japan, 13–16 May, 1981), eds. M. Jimbo, T. Miwa, World Sci., Singapore, 1983, 39–119 | MR | Zbl

[2] I. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1985 | MR | Zbl

[3] U. Fulton, Dzh. Kharris, Teoriya predstavlenii. Nachalnyi kurs, MTsNMO, M., 2017 | MR

[4] T. Miwa, M. Jimbo, E. Date, Solitons. Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge Tracts in Mathematics, 135, Cambridge Univ. Press, Cambridge, 2000 | MR

[5] N. Jing, N. Rozhkovskaya, “Vertex operators arising from Jacobi–Trudi identities”, Commun. Math. Phys., 346:2 (2016), 679–701 | DOI | MR

[6] N. Jing, “Vertex operators and Hall–Littlewood symmetric functions”, Adv. Math., 87:2 (1991), 226–248 | DOI | MR

[7] T. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro, “An elementary introduction to Sato theory”, Prog. Theor. Phys. Suppl., 94 (1988), 210–241 | DOI | MR

[8] K. Takasaki, “Initial value problem for the Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (University of Tokyo, 20–27 December, 1982), Advanced Studies in Pure Mathematics, 4, ed. K. Okamoto, North-Holland, Amsterdam, 1984, 139–163 | DOI | MR | Zbl

[9] T. Takebe, “Representation theoretical meaning of initial value problem for the Toda lattice hierarchy: I”, Lett. Math. Phys., 21:1 (1991), 77–84 | DOI | MR

[10] Dzh. Kharnad, I. V. van de Ler, A. Yu. Orlov, “Kratnye summy i integraly kak tau-funktsii neitralnoi ierarkhii Kadomtseva–Petviashvili”, TMF, 168:1 (2011), 112–124 | DOI | MR

[11] A. Yu. Orlov, T. Shiota, K. Takasaki, Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions, arXiv: 1201.4518

[12] J. W. van de Leur, A. Yu. Orlov, T. Shiota, “CKP hierarchy, bosonic tau function and bosonization formulae”, SIGMA, 8 (2012), 036, arXiv: 1102.0087 | MR | Zbl

[13] J. W. van de Leur, A. Yu. Orlov, “Character expansion of matrix integrals”, J. Phys. A: Math. Theor., 51:2 (2017), 025208, 34 pp. | DOI | MR

[14] B. Fauser, P. D. Jarvis, R. C. King, “Plethysms, replicated Schur functions and series, with applications to vertex operators”, J. Phys. A: Math. Theor., 43:40 (2010), 405202, 30 pp. | DOI | MR

[15] G. Veil, Klassicheskie gruppy, ikh invariantny i predstavleniya, IL, M., 1947 | MR | Zbl

[16] D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford Univ. Press, Oxford, 1958 | MR

[17] B. Fauser, P. D. Jarvis, R. C. King, “Plethystic vertex operators and boson-fermion correspondences”, J. Phys. A: Math. Theor., 49:42 (2016), 425201, 24 pp. | DOI | MR

[18] N. Wang, C. Li, “$\pi$-Type fermions and $\pi$-type KP hierarchy”, Glasg. Math. J., 61:3 (2019), 601–613 | DOI | MR

[19] K. Koike, “On the decomposition of tensor products of the representations of classical groups: by means of universal characters”, Adv. Math., 74:1 (1989), 57–86 | DOI | MR | Zbl

[20] T. Tsuda, “Universal characters and an extension of the KP hierarchy”, Commun. Math. Phys., 248:3 (2004), 501–526 | DOI | MR

[21] T. Tsuda, “From KP/UC hierarchies to Painlevé equations”, Internat. J. Math., 23:5 (2012), 1250010, 59 pp., arXiv: 1004.1347 | DOI | MR

[22] T. Tsuda, “Universal characters, integrable chains and the Painlevé equations”, Adv. Math., 197:2 (2005), 587–606 | DOI | MR

[23] T. Tsuda, “Universal characters and $q$-Painlevé systems”, Commun. Math. Phys., 260:1 (2005), 59–73 | DOI | MR

[24] T. Tsuda, “Universal character and $q$-difference Painlevé equations”, Math. Ann., 345:2 (2009), 395–415 | DOI | MR

[25] Chuan-Chzhun Li, “Silno svyazannye universalnye kharaktery i ierarkhii tipa B”, TMF, 201:3 (2019), 371–381 | DOI | DOI | MR

[26] N. Wang, C. Li, “Universal character, phase model and topological strings on $\mathbb{C}^3$”, Eur. Phys. J. C, 79:11 (2019), 953, 9 pp. | DOI

[27] Chuan-Chzhun Li, “Konechnomernye tau-funktsii ierarkhii universalnykh kharakterov”, TMF, 206:3 (2021), 368–383 | DOI | DOI | MR