Abstract
We suggest an ansatz for representation of affine Yangian \( Y\left({\hat{\mathfrak{gl}}}_1\right) \) by differential operators in the triangular set of time-variables Pa,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting “cut-and-join” generators ψn of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods.
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References
I. Macdonald, Symmetric functions and Hall polynomials, in Oxford Mathematical Monographs, Oxford University Press (1995).
A. Mironov, A. Morozov and S. Natanzon, Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory, Theor. Math. Phys. 166 (2011) 1 [arXiv:0904.4227] [INSPIRE].
C.N. Pope, X. Shen and L.J. Romans, W∞ and the Racah-Wigner Algebra, Nucl. Phys. B 339 (1990) 191 [INSPIRE].
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Representation theory of the W1+∞ algebra, Prog. Theor. Phys. Suppl. 118 (1995) 343 [hep-th/9408158] [INSPIRE].
A. Tsymbaliuk, The affine Yangian of gl1 revisited, Adv. Math. 304 (2017) 583 [arXiv:1404.5240] [INSPIRE].
D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
T. Procházka, W-symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
D. Galakhov, W. Li and M. Yamazaki, Shifted quiver Yangians and representations from BPS crystals, JHEP 08 (2021) 146 [arXiv:2106.01230] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Representations of quantum toroidal gln, J. Algebra 380 (2013) 78.
G. Noshita and A. Watanabe, Shifted quiver quantum toroidal algebra and subcrystal representations, JHEP 05 (2022) 122 [arXiv:2109.02045] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
M. Ghoneim, C. Kozçaz, K. Kurşun and Y. Zenkevich, 4d Higgsed network calculus and elliptic DIM algebra, Nucl. Phys. B 978 (2022) 115740 [arXiv:2012.15352] [INSPIRE].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, The MacMahon R-matrix, JHEP 04 (2019) 097 [arXiv:1810.07676] [INSPIRE].
A. Morozov, Integrability and matrix models, Phys. Usp. 37 (1994) 1 [hep-th/9303139] [INSPIRE].
A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Matrix models of 2D gravity and Toda theory, Nucl. Phys. B 357 (1991) 565 [INSPIRE].
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Towards unified theory of 2d gravity, Nucl. Phys. B 380 (1992) 181 [hep-th/9201013] [INSPIRE].
V.A. Kazakov, I.K. Kostov and N.A. Nekrasov, D particles, matrix integrals and KP hierarchy, Nucl. Phys. B 557 (1999) 413 [hep-th/9810035] [INSPIRE].
A. Morozov, Integrability and Matrix Models, arXiv:2212.02632 [INSPIRE].
F. Liu et al., (q, t)-deformed (skew) Hurwitz τ -functions, Nucl. Phys. B 993 (2023) 116283 [arXiv:2303.00552] [INSPIRE].
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov and W.-Z. Zhao, On KP-integrable skew Hurwitz τ-functions and their β-deformations, Phys. Lett. B 839 (2023) 137805 [arXiv:2301.11877] [INSPIRE].
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, R. Wang and W.-Z. Zhao, Interpolating matrix models for WLZZ series, Eur. Phys. J. C 83 (2023) 377 [arXiv:2301.04107] [INSPIRE].
A. Morozov, Unitary Integrals and Related Matrix Models, Theor. Math. Phys. 162 (2010) 1 [arXiv:0906.3518] [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. Part II. Fundamental representation. Up to five strands in braid, JHEP 03 (2012) 034 [arXiv:1112.2654] [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. Part I. Integrability and difference equations, in Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer, A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger eds., World Scientific (2011), pp. 101–118 [https://doi.org/10.1142/9789814412551_0003] [arXiv:1112.5754] [INSPIRE].
H. Itoyama, A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. Part III. All 3-Strand braids in the first symmetric representation, Int. J. Mod. Phys. A 27 (2012) 1250099 [arXiv:1204.4785] [INSPIRE].
H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett. B 347 (1995) 49 [hep-th/9411053] [INSPIRE].
A. Mironov and A. Morozov, Many-body integrable systems implied by WLZZ models, Phys. Lett. B 842 (2023) 137964 [arXiv:2303.05273] [INSPIRE].
H. Awata, Hidden Algebraic Structure of the Calogero-Sutherland Model, Integral Formula for Jack Polynomial and Their Relativistic Analog, in CRM Series in Mathematical Physics, Springer, New York, NY, U.S.A. (2000), pp. 23–35 [https://doi.org/10.1007/978-1-4612-1206-5_2].
A. Mironov, A. Morozov and Y. Zenkevich, Duality in elliptic Ruijsenaars system and elliptic symmetric functions, Eur. Phys. J. C 81 (2021) 461 [arXiv:2103.02508] [INSPIRE].
A. Mironov and A. Morozov, On Hamiltonians for Kerov functions, Eur. Phys. J. C 80 (2020) 277 [arXiv:1908.05176] [INSPIRE].
A. Mironov and A. Morozov, Superintegrability summary, Phys. Lett. B 835 (2022) 137573 [arXiv:2201.12917] [INSPIRE].
A. Mironov and A. Morozov, Bilinear character correlators in superintegrable theory, Eur. Phys. J. C 83 (2023) 71 [arXiv:2206.02045] [INSPIRE].
A. Mironov and A. Morozov, Superintegrability as the hidden origin of the Nekrasov calculus, Phys. Rev. D 106 (2022) 126004 [arXiv:2207.08242] [INSPIRE].
A. Mironov and A. Morozov, Superintegrability of Kontsevich matrix model, Eur. Phys. J. C 81 (2021) 270 [arXiv:2011.12917] [INSPIRE].
A. Mironov and A. Morozov, On the complete perturbative solution of one-matrix models, Phys. Lett. B 771 (2017) 503 [arXiv:1705.00976] [INSPIRE].
A. Mironov, A. Morozov and Z. Zakirova, New insights into superintegrability from unitary matrix models, Phys. Lett. B 831 (2022) 137178 [arXiv:2203.03869] [INSPIRE].
A. Mironov, V. Mishnyakov, A. Morozov and A. Zhabin, Natanzon-Orlov model and refined superintegrability, Phys. Lett. B 829 (2022) 137041 [arXiv:2112.11371] [INSPIRE].
V. Mishnyakov and A. Oreshina, Superintegrability in β-deformed Gaussian Hermitian matrix model from W-operators, Eur. Phys. J. C 82 (2022) 548 [arXiv:2203.15675] [INSPIRE].
R. Wang, F. Liu, C.-H. Zhang and W.-Z. Zhao, Superintegrability for (β-deformed) partition function hierarchies with W-representations, Eur. Phys. J. C 82 (2022) 902 [arXiv:2206.13038] [INSPIRE].
A. Bawane, P. Karimi and P. Sułkowski, Proving superintegrability in β-deformed eigenvalue models, SciPost Phys. 13 (2022) 069 [arXiv:2206.14763] [INSPIRE].
L. Cassia, R. Lodin and M. Zabzine, On matrix models and their q-deformations, JHEP 10 (2020) 126 [arXiv:2007.10354] [INSPIRE].
A. Morozov, Cauchy formula and the character ring, Eur. Phys. J. C 79 (2019) 76 [arXiv:1812.03853] [INSPIRE].
A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [INSPIRE].
D. Leites and E. Poletaeva, Defining relations for classical Lie algebras of polynomial vector fields, math/0510019.
A. Morozov, M. Reva, N. Tselousov and Y. Zenkevich, Irreducible representations of simple Lie algebras by differential operators, Eur. Phys. J. C 81 (2021) 898 [arXiv:2106.03638] [INSPIRE].
A. Morozov, M. Reva, N. Tselousov and Y. Zenkevich, Polynomial representations of classical Lie algebras and flag varieties, Phys. Lett. B 831 (2022) 137193 [arXiv:2202.11683] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, On elementary proof of AGT relations from six dimensions, Phys. Lett. B 756 (2016) 208 [arXiv:1512.06701] [INSPIRE].
S.V. Kerov, Hall-Littlewood functions and orthogonal polynomials, Funct. Anal. Appl. 25 (1991) 65.
Y. Zenkevich, 3d field theory, plane partitions and triple Macdonald polynomials, JHEP 06 (2019) 012 [arXiv:1712.10300] [INSPIRE].
H. Awata, H. Kanno, A. Mironov and A. Morozov, Shiraishi functor and non-Kerov deformation of Macdonald polynomials, Eur. Phys. J. C 80 (2020) 994 [arXiv:2002.12746] [INSPIRE].
H. Awata, H. Kanno, A. Mironov and A. Morozov, Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function, JHEP 08 (2020) 150 [arXiv:2005.10563] [INSPIRE].
A. Morozov, Cut-and-join operators and Macdonald polynomials from the 3-Schur functions, Theor. Math. Phys. 200 (2019) 938 [arXiv:1810.00395] [INSPIRE].
A. Morozov, An analogue of Schur functions for the plane partitions, Phys. Lett. B 785 (2018) 175 [arXiv:1808.01059] [INSPIRE].
A. Morozov and N. Tselousov, Hunt for 3-Schur polynomials, Phys. Lett. B 840 (2023) 137887 [arXiv:2211.14956] [INSPIRE].
N. Wang, Affine Yangian and 3-Schur functions, Nucl. Phys. B 960 (2020) 115173 [INSPIRE].
Z. Cui, Y. Bai, N. Wang and K. Wu, Jack polynomials and affine Yangian, Nucl. Phys. B 984 (2022) 115986 [INSPIRE].
N. Wang and L. Shi, Affine Yangian and schur functions on plane partitions of 4, J. Math. Phys. 62 (2021) 061701.
Acknowledgments
We are indebted for interesting discussions to S. Barakin, K. Gubarev, K. Khmelevsky, N. Kolganov, E. Lanina, A. Mironov, V. Mishnyakov, E. Musaev, P. Suprun, M. Tsarkov and especially for numerous explanations to D. Galakhov.
Our work is supported by the Russian Science Foundation (Grant No. 20-71-10073).
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Morozov, A., Tselousov, N. 3-Schurs from explicit representation of Yangian \( \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) \). Levels 1–5. J. High Energ. Phys. 2023, 165 (2023). https://doi.org/10.1007/JHEP11(2023)165
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DOI: https://doi.org/10.1007/JHEP11(2023)165