Abstract
For each partition \(\underline{p}\) of an integer \(N\ge 2\), consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the \(\underline{p}\)-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical \(\mathcal {W}\)-algebra \(\mathcal {W}(\mathfrak {gl}_N,\underline{p})\), and write down explicit formulas for evolution of these generators along the commuting Hamiltonian flows.
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Acknowledgements
The first, second, third and fourth author are extremely grateful to the IHES for their kind hospitality during the summer of 2019, when the paper was completed. The first author was supported by a Junior Fellow award from the Simons Foundation. The second author was partially supported by the national PRIN fund n. 2015ZWST2C\(\_\)001 and the University funds n. RM116154CB35DFD3 and RM11715C7FB74D63. The third author is supported by the Bert and Ann Kostant fund. The fourth author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT Grant agreement 677368).
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Communicated by Y. Kawahigashi
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Carpentier, S., De Sole, A., Kac, V.G. et al. \(\underline{p}\)-reduced Multicomponent KP Hierarchy and Classical \(\mathcal {W}\)-algebras \(\mathcal {W}(\mathfrak {gl}_N,\underline{p})\). Commun. Math. Phys. 380, 655–722 (2020). https://doi.org/10.1007/s00220-020-03817-x
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DOI: https://doi.org/10.1007/s00220-020-03817-x