We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative functionals of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to finite temperature Schur measures, and extends both the result of Okounkov in {okounkovschurmeasures} and of Cafasso-Ruzza in {cafassoruzza} concerning the finite-temperature Plancherel measure. Our proof lies on the semi-infinite wedge formalism and the Boson-Fermion correspondance.
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Pierre Lazag (2024) studied this question.
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